Geometric Group Theory Down Under: Proceedings of a Special Year in Geometric Group Theory, Canberra, Australia, 1996 9783110806861, 9783110163667

Table of contents :
The topology of polynomial varieties
Finiteness length and connectivity length for groups
Convergence groups and configuration spaces
Groups, semigroups and finite presentations
Conformal modulus: the graph paper invariant or the conformal shape of an algorithm
Injectivity of the positive monoid for some infinite type Artin groups
Injective maps between Artin groups
The intersection of flat subsets of a braid group
Reconstructing simple group actions
Intersection multiplicities and reflection subquotients of unitary reflection groups I
Some examples of hyperbolic groups
Embedding free amalgams of discrete groups in non-discrete topological groups
Indiscrete representations, laminations, and tilings
Automatic structures on central extensions
Distortion functions for subgroups
Amenable groups, isoperimetric profiles and random walks
Whitehead graphs on handlebodies
List of Contributors
List of Participants

Citation preview

Geometrie Group Theory Down Under

1749

1999

Geometric Group Theory Down Under Proceedings of a Special Year in Geometric Group Theory, Canberra, Australia, 1996

Editors John Cossey Charles F. Miller III Walter D. Neumann Michael Shapiro

w DE

G Walter de Gruyter · Berlin · New York 1999

John Cossey Math. Department School of Math. Sciences Australian National University Canberra, ACT 0200 Australia

Editors Charles F. Miller III, Walter D. Neumann Dept. of Math. & Statistics University of Melbourne Parkville, Victoria 3052 Australia

Michael Shapiro Dept. of Math. & Computer Science Rutgers University Newark, NJ USA

7997 Mathematics Subject Classification: 20-XX Keywords: Geometric group theory, presentation, representation, graph, group action, Artin group, braid group, automatic group, hyperbolic group

1

Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability. Library of Congress — Cataloging-in-Publication-Data Geometric group theory down under : proceedings of a special year in geometric group theory, Canberra, Australia, 1996 / editors, John Cossey ... [et al.]. p. cm. Papers from the International Conference on Geometric Group Theory held in Canberra from 14-19 July, 1996. ISBN 3-11-016366-7 (alk. paper) 1. Geometric group theory - Congresses. I. Cossey, John, 1941- . II. International Conference on Geometric Group Theory (1996 : Canberra, A.C.T.) QA183.G462 1999 512'.2-dc21 98-53281 CIP

Die Deutsche Bibliothek — Cataloging-in-Publication-Data Geometric group theory down under : proceedings of a special year in geometric group theory, Canberra, Australia, 1996 / ed. John Cossey ... - Berlin ; New York : de Gruyter, 1999 ISBN 3-11-016366-7

© Copyright 1999 by Walter de Gruyter GmbH & Co. KG, D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Typeset using the authors' ΤεΧ files: I. Zimmermann, Freiburg. Printing: WB-Druck GmbH & Co., Rieden/Allg u. Binding: L deritz & Bauer, Berlin. Cover design: Thomas Bonnie, Hamburg.

Preface The International Conference on Geometric Group Theory was held in Canberra from 14-19 July, 1996. It was a high point of a Special Year in Geometric Group Theory sponsored by the School of Mathematical Sciences of the Australian National University. The program of the Special Year also included a workshop for graduate students from 22 January to 9 February, a number of related conferences and an active visitor program throughout the year. The organisation was concentrated at ANU and the University of Melbourne, but many other Australian and New Zealand universities made valuable contributions. The conference attracted 84 participants: 41 from Australia, 3 from New Zealand and 40 from other countries. The theme of Geometric Group Theory was interpreted quite broadly to include the various aspects of the interplay between geometry and group theory. A program of 25 invited talks covered a wide range of topics. This volume represents the efforts of all who contributed papers for it and the selfless efforts of the referees. We are grateful to all of them. We are also grateful for the assistance of Beata Wysocka for editing the submissions into the de Gruyter TgX style. The smooth running of a conference depends greatly on the quality of administrative support. Our warmest thanks to all who helped, especially to the administrative staff of the Centre for Mathematics and its Applications of the ANU, of Ursula College in Canberra (where most participants stayed), and of the Department of Mathematics of Melbourne University for their support. The Special Year was funded initially by a grant from the School of Mathematical Sciences of the ANU. This enabled initial planning to be undertaken with some confidence. Substantial contributions were made by several other universities in Australia and New Zealand by providing support for their own members as well as helping to support overseas visitors. Melbourne University in particular supported a large number of overseas visitors. Much of the visitor support was provided within projects funded by Australian Research Council. The Australian Mathematical Society also contributed towards the cost of the Conference.

Conference Program Sunday 9.45-11.00

G. Baumslag: Reflections on algebraic geometry over groups

11.30-12.30 G.Martin: The geometry of Kleinian groups 2.00-3.00 S. Smith: Applications of finite geometries to group cohomology 3.30-4.00

C. Pittet: Isoperimetric profiles of solvable groups

4.00-4.30

J. Crisp: Injectivity of maps between Artin groups

4.30-5.30

L. Mosher: Quasi-isometry classification of the solvable Baumslag-Solitar groups Drinks and discussion at Ursula College Bar

7.00 Monday 9.00-10.00

M. Gromov: On the group geometry related to the Novikov conjecture 10.00-11.00 C. Leedham-Green: Structure and classification of p-groups and pro-p-groups 11.30-12.30 B.Bowditch: Connectedness properties of ideal boundaries and limit sets 2.00-3.00 V. Obraztsov: Some new embedding constructions 3.30-4.00 J. Hudson: Canonical words for braids 4.00-4.30 G. Willis: Totally disconnected, locally compact groups 4.30-5.30 H. Short: Isoperimetric inequalities for subgroups of HNN extensions Tuesday 9.00-10.00

J. Stalling*: Cut vertices and waves 10.00-11.00 M.Conder: Group actions and regular maps on non-orientable surfaces

viii

Conference Program

11.30-12.30 R. Chamey: Nonpositively curved piecewise Euclidean structures on hyperbolic manifolds 2.00-3.00 3.30-4.00 4.00-4.30 4.30-5.30

E. O'Brien: Tensor factorisations of matrix groups R. Jiang: Out(F„)-action on a simplicial complex S. Pride: The geometry of group extensions M. Shapiro: Almost a generalization of a theorem of Muller and Schupp

Wednesday 9.00-10.00

Z. Sela: Moduli spaces of residually free groups 10.00-10.30 R. Oilman: A small cancellation characterization of hyperbolic groups 10.30-11.00 S. Matsumoto: Subgroup separability of 3-manifold groups 11.30-12.30 C.Praeger: The number of fixed-point-free orthogonal transformations of finite vector spaces of even order 1.30 Bus for Excursion leaves from Ursula College

Thursday 9.00-10.00

A. Ol'shanskii: Locally finite subgroups of free Burnside groups of large even exponents 10.00-11.00 /. G. Macdonald: Affine Hecke Algebras and orthogonal polynomials 11.30-12.30 G. Swamp: Tracks and splittings of groups 2.00-3.00 W. Kantor: Reconstructing classical group actions 3.30-4.00 S. Hermiller: Artin groups, rewriting systems and three manifolds 4.00-4.30 4.30-5.30

A. C. Kim: A group presentation and 3-dimensional manifolds W. Neumann: Biautomaticity of central extensions and bounded cohomology

Conference Program

ix

Friday 9.00-10.00

J. Cannon: On the conformality of planar tiling-sequences

10.00-10.30 S.Glasby: Unique tensor factorization of irreducible projective representations of groups 10.30-11.00 D. Cooper: Actions on rational homology 3-spheres 11.30-12.30 G. Lehrer: Invariant hypersurfaces and reflection groups 2.00-3.00

R. Bieri: Openness results for group actions on metric spaces

3.30-4.30

M. Bridson: Groups acting on non-positively curved complexes

Table of Contents Barbu Berceanu The topology of polynomial varieties

1

Robert Bieri Finiteness length and connectivity length for groups

9

B. H. Bowditch Convergence groups and configuration spaces

23

Colin M. Campbell, Edmund F. Robertson, Nikola Ruskuc, and Richard M. Thomas Groups, semigroups and finite presentations

55

J. W. Cannon, W. J. Floyd, and W. R. Parry Conformal modulus: the graph paper invariant or the conformal shape of an algorithm

71

Ruth Charney Injectivity of the positive monoid for some infinite type Artin groups

103

John Crisp Injective maps between Artin groups

119

John F. P. Hudson The intersection of flat subsets of a braid group

139

William M. Kantor and and Tim Penttila Reconstructing simple group actions

147

G. I. Lehrer and T. A. Springer Intersection multiplicities and reflection subquotients of unitary reflection groups I

181

C. F. Miller III, Walter D. Neumann and G. A. Swarup Some examples of hyperbolic groups

195

Sidney A. Morris and Viatcheslav N. Obraztsov Embedding free amalgams of discrete groups in non-discrete topological groups

203

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Table of Contents

Lee Mosher Indiscrete representations, laminations, and tilings

225

Walter D. Neumann and Michael Shapiro Automatic structures on central extensions

261

A. Yu. Ol'shanskii Distortion functions for subgroups

281

Christophe Pittet and Laurent Saloff-Coste Amenable groups, isoperimetric profiles and random walks

293

John R. Stallings Whitehead graphs on handlebodies

317

List of Contributors

331

List of Participants

333

The topology of polynomial varieties Barbu Berceanu

Abstract. We show that the affine Cremona group G(n) has the homotopy type of the general linear group GL„ (C) and that two closed related spaces, the group of elementary transformations and the space of polynomial transformations with non-zero jacobian, have the same homotopy type; we introduce the polynomial varieties as spaces of polynomial solutions of algebraic equations and show that they have the homotopy type of a finite CW-complex. 1991 Mathematics Subject Classification: 14E07. Key words and phrases: affine Cremona group, homotopy type, polynomial equations.

1. Our first aim is to describe the homotopy type of the group of algebraic automorphisms of the affine space C". Let A = C [ X i , . . . , Xn] be the ring of polynomials in n variables. We call F = (Fi , . . . , Frt) e An an algebraic automorphism if there is a G e A" such that F o G = id, G o F = id (as we restrict ourselves to the complex field, F e A" bijective implies F"1 € A" ). The group of these automorphisms will be denoted by C(n), the n-th Cremona group. Thesubspace A^ = {/ € A | deg(/) < d] has a canonical vector space topology and we shall consider the direct limit topology

On An we will consider the product topology, which is the same as lim^ [A(d)]n (the inclusions A^ —> A^ +1) are closed). The topology on the Cremona group is the induced topology and this will be the case with any set of polynomials S C Ak we will consider. Let us remark that the topology on S is given by lim_+ S^d\ where S(d) = S(~][A(d)]k. A map /: S -» T between two set of polynomials S C A*, T C Ah is called a degree map ( a α , ) ρ ι ^ 2 ,α 2ι1 · · · ^*,α*,Λ

7^ Ο to be o,· (m) = min |a//j|

The topology of polynomial varieties

7

(if DjiU are missing in m, then Oj(m) = oo). If ο = (p\, . . . , on) e N" is fixed, we define the degree of Μ relative to ο by n

k

Pj

deg0(m) = J^flf - 5D5I^,A(|ay,Al - o/(wt)). j=l h=\

j= l

Consider a nonzero differential polynomial E e < D Wi ^; its j -order is 0j(£) = min{0/(m) | m monomial with nonzero coefficient in E}. Definition 5. E in£>n,k is said to be homogeneous if all the monomials occuring in E have the same degree relative to o(E) = (o\(E), . . . , on(E)). An ideal D e R are equivalent when restricted to the m-skeleton Xm, in the sense that there is a constant C (dependant on h and h') with \h(x) - h'(x)\ < C for all x e Xm. 'This is not the whole truth: The first ("1-dimensional") invariant of [BS1] and [BNS] was introduced in order to express a new necessary condition for the finitely generated group G to be finitely presented.

Finiteness length and connectivity length for groups

11

Proof. The construction of h is inductively, skeleton by skeleton, using only the facts that X is a G-free complex and R is contactible. (ii) is immediate from the fmiteness assumption on Xm. D We say that X is η-connected in direction +00, if there is a constant 'lag' L e R with the property that for each k < n all sphere maps f:Sk^-X have an extension /: B*+1 -> X with infhf(Mk+[)>mfhf(Sk)-L.

(4)

The point here is that the lag L does not depend upon the map /; but it may well depend upon the height function h. However, it is clear from Proposition 2.1 (ii) that the existence of such a lag will not depend on h. In fact, much more is true: Proposition 2.2. For n < m the condition that X be η-connected in direction +00 depends only on the character χ and not on the particular choice ofX and ofh. Proof. (Sketch, see [BR, Rl].) Assume (X, h) is given. An elementary expansion of the complex Υ = G\X in the sense of simple homotopy theory corresponds to a G-orbit of simple expansions of X and leads to a new free G-CW-complex X'. The height function h: X —> R extends to an equivariant height function h': X' —>· R which can be chosen arbitrarily on representatives of the two new cell orbits of X''. By Proposition 2.1 (ii) one observes that if X is «-connected in direction +00, so is X' and vice versa. Proposition 2.2 can now be proven by showing that if Y\, ¥2 are two K(G, 1) complexes with finite m-skeleta then Y\ can be transformed, by finitely many elementary expansions and contractions, into a CW-complex Ύ[ whose m-skeleton is isomorphic to the m -skeleton of ΎΊ. D Proposition 2.2 allows us to define the connectivity length of the character χ by putting cl(x) := sup{m | m < fl G, X is (m — l)-connected in direction + 00}

(5)

Thus, cl is a function Hom(G, R) —»· NOO which is defined for any group G. The finiteness length fl G is an upper bound for cl(x) and this upper bound is achieved for χ = 0. Moreover, if r is a positive real number then r · h can be used as a height function for r · χ, and so (1.1), multiplied by r, shows that we have cl(rx) = cl(x),

for all 0 < r e R.

(6)

The homotopical geometric invariants of [BR] and [Rl, R2] are the full preimages

2.2. The homological theory Under the same assumptions on G, m, X and «: X —> R one can set up a chain version of 1.1 which results in a homological connectivity length function hcl: Hom(G, R) —>

12

Robert Bieri

NOQ. The concept of homological «-connectivity in direction +00 which we need is obtained by replacing the maps (/, /): (B*+1, Sk) —>· X by cellular chains (c, c) with dc = c, and by interpreting the images /(B/c+1) and f(Sk) in (4) as the supports supp(c) and supp(c) in X. Everything works, in fact, under the assumption that X is merely acyclic rather than contractible, so that we need G of type FHm rather than of type Fm. The only difficulty is that there is no homological analogue of elementary expansions and contractions on a space X, so that the analogue of Proposition 2.2 is not immediate. The problem disappears when we pass to the more general algebraic theory and use non-geometric chain homotopies and arbitrary free resolutions.

2.3. The algebraic connectivity length Here we replace the free G-CW-complex X of Section 2.1 by a free resolution F_ of the trivial G-module Z which is finitely generated in all dimensions < in (so that 0 < m < acl G). The role of the height function h : X —>· R is played by a "valuation on the free resolution", v: F_—> R U {00} introduced in [BR]. These valuations mimik the function inf /z(supp(— )) : C —> M U {00} which is available if F_ = C(X) is the cellular chain complex of an acyclic space X. Given the valuation υ we say that F_ is (m — l)-acyclic in the direction +00 if there is a lag L such that for each c € F^, k < m, there is some c e F/t+i with dc = c and

v(c) < v(c) - L

(7)

Proposition 2. 1 and Proposition 2.2 have algebraic counterparts and we end up with the definition of the algebraic connectivity length of a character acl(x) := sup{m | m < afl G, F_ is (m — l)-acyclic in direction + 00} The properties of cl(x) hold, mutatis mutandis, for acl. As an extra bonus we can apply the methods to any free resolution and thus obtain an (algebraic) connectivity length function, acl^, on Hom(G, M) for every G-module A.

2.4. Relations between cl and acl By using the cellular chain complex of a free contractible G-CW-complex X to compute acl(x), one observes readily that cl(x) < acl(x) for all χ e Hom(G, R). Also, it is obvious that cl(x) = 0 if and only if acl(x) = 0 and, moreover, using the Hurewicz Theorem one can prove that if the group G is finitely presented then cl(x) > 2 implies cl(x) = acl(x). So, for finitely presented groups the two functions cl and acl can only disagree on characters of connectivity length 1. Bestvina and Brady have recently shown that for suitable graph groups G(F) one can, indeed, find even discrete characters χ : G (Γ) -» Ζ with cl(x) φ· acl(x). Recall

Finiteness length and connectivity length for groups

13

that if Γ is a finite simplicial graph then G(F) is given by the presentation G(F) = {ver Γ | [v, w] = l

({v, w} e edgD)

Let Κ(Γ) denote the full simplicial complex on Γ, that is, the simplices of Κ(Γ)) are those subsets of ver Γ in which all pairs of elements are joined by an edge of Γ. The main result of [BB] asserts that if χ : G (Γ) -» Ζ is a character with χ (υ) ^ 0 for all ν 6 ver Γ, then cl(x) = cl Κ(Γ) and acl(x) = hcl ΑΓ(Γ); here cl X (resp. hcl X) stands for the (homological) connectivity length of a space X and is defined to be the largest number m with the property that X is (homologically) (m — 1)-connected. It is easy to construct graphs Γ with cl K(Y} φ hcl Κ(Γ): take Γ to be the 1-skeleton of the baricentric subdivision of an acyclic but not 1-connected simplicial complex. J. Meier, H. Meinert and L. VanWyk have recently completed the full computation of cl and acl for graph groups [MMV1].

2.5. Some properties of cl and acl For completeness we mention properties of the geometric invariants E W (G) = cl~'([m, oo]). Everything holds, mutatis mutandis, for acl. (a) E m (G) — {0} is for all groups G and all 0 < m < oo an open (conical) subset of Hom(G, R). This is the basic openness result of [BNS, BR, Rl, R2]. (b) In all known cases E W (G) — {0} is polyhedral, i.e., a finite union of finite intersections of open half spaces. (c) In almost all known cases the open half spaces mentioned in (b) above can be chosen as the closure of rational half spaces in Hom(G, Q). The only known counter examples here are certain groups of PL-homeomorphisms of the unit interval [BNS].

3. Connectivity length versus finiteness length 3.1. The main theorem The main theorem of [BR] and [Rl, R2] translates into a formula which computes the finiteness length functions fl, afl: jV = G(V) -> NOO from the introduction in terms of the connectivity length functions cl, acl: Hom(G, K) = Hom(V, R) -> N^ of Section 2 (recall that V = (G/G') Q). Theorem 3.1. If G is a finitely generated group and N e JJ (i.e. G' < N < G), then fl N = inf{cl(x) | χ € Hom(G, M) with χ(Ν) = 0} and similarly for afl.

(8)

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Robert Bieri

Corollary 3.2. If N is the kernel of a non-zero rational character χ : G —>· Q then

fltf = inf{cl(x),cl(-x)}.

(9)

and similarly for afl. Proof. As G is finitely generated χ (G) is either infinite cyclic or trivial. Hence the subspace (χ' € Hom(G, R) | χ'(Ν} = 0} is one dimensional and thus consist of the three (R>0)-orbits (R> 0 )x, Ο, -(R>o)x· Π

3.2. An application We should like to discuss to what extent the finiteness length functions fl, afl: G(V) —»· NOO are determined by their restriction to the subset Gl(V) = {W \ W < V, dimq(V/W) = 1}. If the group G is finitely generated — and we will always assume this is the case — the elements of G1 (V) are represented by the elements in

Μ :={N εΜ \ G/N = 1}. Let us say that our finitely generated group G satisfies the rational-polyhedrality condition if all its geometric invariants T,m(G} = cl~'([m, oo]) — {0} are rational and polyhedral in the sense of (c) in Section 2.5. Examples of such groups are all metabelian groups of finite rank [M2, M3], all graph groups [MMV1], and all finite direct products of 1-relator and/or 3-manifold-groups. But one can expect the condition to hold in rather greater generality — see Section 2.5 . Corollary 3.3. Assume the finitely generated group G satisfies the rational polyhedrality condition, and let N € «V with G/N infinite. Then fl N = inf {fl M | N < M € M}

(10)

and similarly for afl. Proof. Let m be the right hand side of equation (10). Then, clearly, fiN < m by Theorem 3.1. or by an easy direct argument. Now let W < Hom(G, R) be the linear subspace {χ | χ(ΛΟ = 0}. W is a rational subspace and so W Π E m (G) is a rational polyhedral subset of W. If χ : G —> Q is a rational character of W then, by Corollary 3.2, cl(x) > fl(kerx) > m and so χ e E W (G). This shows that the rational polyhedral subset W D E m (G) contains all rational points of W. By elementary arguments this implies that W Π Σ™ (G) = W, and so cl(x) > m for all χ e W. Hence fl(AO > m by Theorem 3.1. D

Finiteness length and connectivity length for groups

15

4. Direct products In the contextof direct products it is convenient to have modified connectedness length functions cl, acl: Hom(G, E) ->· NOO at hand. They are defined by i cl( X ) + l, cl(x) := · 0

ίίχ ^ 0 if χ = 0

and similarly for acl. Note that both cl(/) = 0 or acl(x) = 0 are equivalent to χ =0.

4.1. The Direct-Product-Conjecture Meinert's computation of the geometric invariant of direct products of free groups [Ml] led to a conjecture which becomes particularly handy when stated in terms of the modified connectivity length functions Conjecture 4.1. 2 ' 3 Let G = G\ χ G^ with G\ and G^_ (and hence G)fo type Then we have

(11) for each character χ = (χ\, χ2) : G —> R. Until recently I had hoped that a similar formula would hold for cl; but Meinert, Meier and VanWyk have pointed out that this would contradict their computation of the geometric invariants of graph groups [MMV1]. In fact, already the BestvinaBrady computation of cl(x) for the generic character χ : G (Γ) —>· Ζ sketched in Section 2.4 yields such a contradiction: the Bestvina-Brady result asserts that cl(x) is the connectivity length cl Κ(Γ) of the flag complex Κ(Γ) (i.e., the full simplicial complex on Γ). Now the direct product of two graph groups G(Fi) χ G(F2) is the graph group G(F) of the graph theoretic join Γ of ΓΙ and Γ2, which is the 1-skeleton ofthejoinArirO*/^!^). Hence Κ(Γ} = Κ ( Γ ι ) * Κ ( Γ 2 ) . Thus the Bestvina-Brady result shows that the behaviour of the connectivity length functions cl(x), acl(x) on the generic character χ with respect to direct products is exactly the same as the behaviour of the connectivity length functions cl(X), hcl(X) with respect to joins of spaces. On the one hand, now, a Maier- Vietoris argument computes the homology of a join X * Υ and one finds that the function (hcl +1) is, indeed, additive with respect to joins. 2

In the language of the geometric invariants the Conjecture has the following form: Let E m (G; Z)c denote the complement of the algebraic invariant in S(G). Then E W (G;Z) C = UPEP(G\;%)C * E m -^(G 2 ; Z) c , where * stands for the join in S(G) = S(G\)* S(G 2 ). Added in proof. Ross Geoghegan and I have now a proof of the "Direct-Product-Conjecture over any field K"; i.e., the join formula in footenote 3) holds true if Σ*(·; Ζ) is replaced by Σ*(·; Κ).

16

Robert Bieri

This is evidence in favour of the conjecture — in fact one can use the Meier-MeinertVanWyk computation [MMV1] in order to show that the Direct-Product-Conjecture does hold for graph groups. On the other hand, if X and Υ are connected then X * Υ is 1-connected and so the Hurewicz Theorem shows cl(X * Y) = hcl(X * Y). This shows that (cl +1) cannot be additive when the spaces X and Υ are 1-acyclic but not 1-connected; indeed, clX = c\Y = I and hc\X = hc\Y > 2, but cl(X * 7) = hcl(X * Y) = hcl X + hcl Υ + I > 5.

4.2. Meinert's inequality The following inequality has its roots in Holger Meinert's diploma thesis [Ml]. Theorem 4.2. IfG = G\x G2, then we have for each character χ — (χ\, χ2) : G^R cl(x) > cl(Xi) + cl(X2) a c ( x ) > a c ( x i ) + ad( X2 )

ifGi, G2 are of type F^

(12)

ifGi,G2 are of type FP^

(13)

Moreover, i f c l ( x \ ) (resp. ac\(x\)) is zero or infinity then equality holds. Proof. A typical application of the "Σ™ -Criterion". This criterion which has its roots in [BS1] and kept reappearing through [BNS, BR, Rl, R2], was the first tool to compute the geometric invariants. My favourite version is the one in [BS2] which, in the homotopical framework, goes as follows: Let G, χ , X and h : X ->· R as in Section 2.1. Then cl(x) > m if and only if there are real numbers ε, μ, with ε > 0, and a cellular deformation ^ : X x [0, oo) —> X satisfying the Lipschitz conditions

if χ € X

>

(14)

for all 11 < t2 in [0, oo). Now we assume that cl(x,) > ra( for ί = 1, 2. Then the Σm'-criterion above holds (all symbols endowed with subscripts i). We use the CW-complex X = X\ χ Χ2 endowed with the obvious G-action and the G-equivariant height function h = h\ +h2 and aim to prove cl(x) > m again using the Em-criterion where m — m\ + m2 + 1. For this we decompose the parameter ray [0, oo) into finite intervals and alternatingly perform the deformations ψ\ χ Id and Id χ τ/^2 on these. In this way we can achieve that (14) holds with ε = min (ει, ε2) and μ = μι + μ2, and with the stronger inequality for all points χ in Xp χ Xq as long as either ρ < m\ or q < m2 — and so certainly for ρ + q < m\ + m2 + 1. This shows that cl(x) > m\ + m2 + 1, and hence, if χ{ φ Ο φ χ2, cl(x) > (mi + 1) + (m2 + 1). The argument breakes down when χ\ or X2 is zero. But if χ2 = 0, say, one can choose h2 = 0. Then the contractible space

Finiteness length and connectivity length for groups

17

X2 is in the fibers of h: X -> E, and so cl(x) = cl(xi) can be seen right from the original definition in Section 2.1. The corresponding results for acl have analogous proofs where the space X is replaced by a free resolution and the cellular deformation •φ· by a chain homotopy. D

4.3. Towards the converse Ross Geoghegan and I have recently proved the following Theorem 4.3. Let G = G\ x G? as in the Conjecture, and assume that G\ and GI (and hence also G) are of type F^. If acl(xi) € {0, 1, 00} then we have

(15) The only new case here is, of course, acl(xi) = 1. We remark that the assumption acl(xi) 6 {0, 1, 00} is always satisfied when G\ is a 1-relator group or a 3-manifold group ([BR]). Therefore, a straightforward induction computes the function acl: Hom(G, R) —> Ν,χ, on the direct product G = G\ x 62 x · · · x Gn of 1-relator and/or 3-manifold groups. This last result is, in fact, due to Ralf Gehrke who showed that certain commutativity assumption on a group G imply vanishing of higher geometric invariants. In particular he had a direct proof of (15) in the case when G = G\ x G^ x · · · x Gn and χ = ( χ ι , . · - , χ«) with acl(x,) = 1 simultaneously for all i, [Gl], [G2]. Corollary 4.4. The Direct-Product-Conjecture holds true if both factors G\ and G^ are direct products of \-relator and/or 3-manifold groups. The proof of Theorem 4.3 will appear elsewhere. Instead, I offer an account on Gehrke's result which is based on but is more transparent than his original proof in [G2]. The key argument is a simple observation on η-cycles in the product P = T\ x · · · x Tn of n R-trees. For each component i = 1, 2 , . . . , η we choose a geodesic path given by an isometric embedding wi: 7, ^> 7/ of an Eucledian intervall 7,. Let w = (w\,... ,wn): K —>· P denote the resulting embedding of the Euclidean brick 7] x · · · x In. Lemma 4.5. Choose a triangulation ofdK and let z € Cn-\(P) denote the singular (n — l)-cycle defined by the embedding dK >—»· P. If c e Cn(P) is any singular η-chain with z = dc then w(K) C supp(c). Proof. Let a = (a\,..., an) be an interior point of K. For each i = 1 , . . . , η we construct a map pi: Γ, —> 7, as follows: we consider the connected component A of Ti — {ΐϋ,·(ο/)} containing u>, (0). Then we put for each t e Γ, max(0, a, — d(t, tu, (a,)), li\,ai+d(t,Wi(ai»,

if t € A, iff g A,

18

Robert Bieri

where d: Τ/ χ 7} —»· R stands for the metric of 7). Note that if t = if,· (&), for some b e //, then d(t, if, (a,)) = \b — «,· |, and so p(wi(b)) = b. Thus p( is a retraction of if;. The crucial piont is that pi is continuous and that its fiber over a, is the singleton set {if (a,·)}. Let/o: Ρ —>· AT be the product map. Again, ρ is a continuous retraction of if: Κ —> Ρ with the singleton fiber {if (α)}. The restrictions if |3/sT and/o|(P —{if (0)}) compose to the embedding dK >—> Κ — {0} which induces an isomorphism in homology. As this isomorphism factors via Hn-\(dK) ->· Hn-\(P — {if(fl)}) it becomes obvious that ζ is homologous to 0 in Ρ — {if (α)}. Hence Ρ - {if (α)} does not support a chain c with 3c = ζ. Π ^ The proof of Gehrke's theorem (i.e., that aclx,· = 1, for ι = 1 , . . . , η implies aclx < n) can now be completed as follows. We use Ken Brown's description [B] of the invariant Σ ι (Gt·) in terms of R-tree actions. A good way to express that result is by saying that aclx, = 1 if an only if the height function A,·: X,· —> R factors via G ;-maps V Λ,·

K

\> Τ li

λ

'\>TO K,

with (7}, λ,) a rooted G, — R-tree which has no G, -invariant line (see [BS2] or [G2]). That (Ti, λ ( ) is rooted means that each point f e 7} is the origin of a unique geodesic ray Rt and λ, maps Rt isometrically onto (—00, λ,·(ί)]· Using the absence of G,-invariant lines one finds two hyperbolic elements x, y e G, whose axes Ax, Ay in 71, do not coincide. We can then take the 1-skeleton of X,· to be the Cayley graph of G/ with respect to a generating set containing jc and y; and then it is easy to find an edge path if,: /, —»· X?, parametrized on / = [0, £,·], such that KIWI is a geodesic path in 7} which runs along Ax U Ay and satisfies inf h,-Wi(d/,·) - inf h, w,·(It·) > r,

(16)

for an arbitrary given real number r. Let K = I\ χ · · · χ In, Ρ = TI χ · · · χ Tn, Χ = Χ\ χ · · · χ X„, let if: Κ —>· Χ, κ: Χ —>· Ρ be the product maps and λ: Ρ -» R given by summation. Then h — λκ is a G-equivariant height function and (16)yields mfhw(dK) - inf hw(K) > r.

(17)

Let now ζ be the (n — 1)cycle of X given by if: dK >—+ X, and let c be any «-chain of X with dc = z. Then we can apply Lemma 4.5 to the cycle κ*(ζ) Ε C n _i (Ρ) and the chain *:*( inf/i(supp(c)) so that inf /i(supp(z)) — inf/i(supp(c)) > r has no lower bound independent of z. This shows that (7) fails to hold, and so acl(x) < η — 1, as asserted.

Finiteness length and connectivity length for groups

19

4.4. The homotopical connectivity length There is no need to search for a separate formula that would compute the homotopical connectivity length cl on a direct product, because Meier, Meinen and VanWyk have observed in [MMV2]: Proposition 4.6. Let G = G\ χ G^, with G \ and G^_ of type F^, and /ei χ = (χ ι, Χ2) be a character ofG. Then acl(x),

ifx\

cl(x) = cl(xi)+cl(x2),

otherwise.

Proof. If one of χι or χ2 is zero Theorem 4.2 applies. If cl(xi) + cl(x2) > 3 then, by Meinert's inequality cl(x) > 3 and so cl(x) = acl(x) by Section 2.4. So it remains to consider the case cl(xi) = 1 = cl(x2). But then Theorem 4.3 applies; hence 2 > aci(x) > cl(x) > ci(xi) + ci( X2 ) = 2.

D

5. Groups with antipodal symmetry 5.1. Let us say thatagroup G has the antipodal symmetry if cl(x) = cl(—χ) for every character χ : G —>· R. Similarly we can talk about the algebraic antipodal symmetry. Examples of groups which are known to have homotopical and algebraic antipodal symmetry are all 3-manifold groups [BNS] and all graph groups [MMV1]. The Direct-Product-Conjecture of Section 4.1 would imply that the antipodal symmetry is preserved under direct products. For the time being we know at least (either by Theorem 4.2 and Theorem 4.3 or by Gehrke's Theorem) that G = G \ χ G2 χ · · · x G„ has homotopical and algebraic antipodal symmetry if all factors are finitely generated 3-manifold groups. In contrast to the rational polyhedrality property of Section 3.2, groups which do not have the antipodal symmetry are easy to find: some of the most simple minded 1-relator groups like G = (χ, a \ xax~l = a2) do not have it. Indeed, if χ : G —>· R is the character with χ(χ) = 1 then cl(x) = 0, but cl(—χ) = oo. 5.2. Parallel to acl we define the function aft G : G(V) -> Ν by putting, for N e Jf aflW+l,

if\G:N\

aflc(tf) = 0,

Then Corollary 3.2 in conjunction with the Direct-Product-Conjecture computes aflc(AO for Ν < G = G\ χ G2. This is particularly simple in the case when

20

Robert Bieri

M = ker(x), with χ = (χι , χ 2 ) a rational character χ : G -»· Q, where we find aflc(Af) = inf (ad(xi) + ad( X2 ), acl(-/i) + ad(-x2)).

(18)

It_is ratherjnteresting to observe that if at least for one of ι = 1 or 2 we have acl(x,) = acl(— χ,·) then the right hand side of (18) is equal to inf (acl(xi), acl(-/i)) + inf (ad(X2), aci(-/2)),

(19)

so that we obtain, by Corollary 3.2, aflc(M) = a f l G l ( G i n M ) + afl G 2 (G 2 nM).

(20)

Conversely, let us assume that acl(x,) φ acl(^x/) both for i = 1 and 2. Replacing χ, by —χ/, if necessary, we may then assume acl(x,) < acl(—_Xj) for / = 1,2. Then χ, φ 0 and we know, from Corollary 3.2, that aflc, (ker χ,) — acl(x,). We find, using Corollary 3.2 and Theorem 4.2, afl c (ker(xi, -χ 2 )) = inf(acl(xi, -χ2), acl(-xi, χ 2 )) > inf (acl(xi) + acl(-x2), acl(-xi) + acl(/2)) acl(x2), whereas aflG,(ker(xi)) + aflc 2 (ker(— χ 2 )) = acl(xi) + acl(x2). This shows that formula (20) fails for Μ — ker(xi , — χ 2 ). We summarize Theorem 5.1. Let G = G\ χ G 2 with both G\ and G 2 of type FPoo. Then formula (20) can only hold true for all M < G with G/M ~ Z if one ofG\ or G 2 has the algebraic antipodal symmetry with respect to rational characters. Conversely, if we assume that one ofG\ or G 2 does have this symmetry then we have afl c (M) > afl G l (Gi n M ) + aflc 2 (G 2 nA/)

(21)

for each M < G with G/M = Z. In a situation when the Direct-Product-Conjecture holds true (e.g. when G\ and G2 are products of \-relator and/or 3-manifold groups then (21) is actually an equality. The following application contains Holger Meinert's computation of afl of a direct product of virtually free group [Ml] as a special case — though, for simplicity, we restrict attention to G/M = Z. Corollary 5.2. Let G — G \ χ · · · χ Gn be a direct product of finitely generated 3-manifold groups, and Μ < G with G/M = Z. Then fl M = afl M = {

oo,

ifGi Π Μ is f. g. and φ G, for some i

#{i | G , g M} - 1, otherwise.

Finiteness length and connectivity length for groups

21

Proof. 3-manifold groups have the antipodal symmetry and satisfy acl(Hom(G,, R)) c {0, 1, 00}. Hence Theorem 5.1 applies, and we find, by induction, that fie M = aflcM is the sum of all aflG,.(G,· Π M). If G, Π M has finite index in G, then G, C Λ/. If | G, : (G, Π Μ) Ι = oo and G, Π Μ isfinitelygenerated, then by Peter Soctt's Theorem, G, Π Μ is also finitely presented and, in fact, of type FOQ, so that aclc, (G; Π Λ/) = oo. In the remaining case aflc, (G, Π A/) = 1, whence the result. D 5.3. acl versus afl again. Theorem 4.1 shows once again the clear advantage of the function acl: Hom(G, R) -> NOO over afl: G(V) -> N^: whereas acl can be expected to be additive with respect to the direct product, the corresponding additivity for afl will only hold true under the assumption of antipodal symmetry. On the other hand it turns out that the finiteness length function afl: G( V) —> NOO considered on suitable direct products G ι χ G does, to some extent, determine the connectivity length function acl: Hom(G, R) -»· NOO. For G j we can take the group GI = (a, x | xax~^ = a1}. We put χ ι : G\ -> Q to be the character given by Xi(jt) = 1 and prove Corollary 5.3. Let G be a group of type FQQ and / : G —> G a rational character. If Μ = ker((xi, χ): G] χ G -» Q) then we have acl(x) = afl M. Proof. By Corollary 3.2, afl(M) is the inf of {acl(xi, χ), acl(-xi, -χ)}; but the second number is, by Meinert's inequality > acl(—χι) + acl(—χ) which is = oo since acl(—χι) = oo. This shows that afl Μ = acl(xi, χ) = acl(xi) + acl(x) — 1, by Theorem 4.3. Since acl(xi) = 1, the corollary follows. D Corollary 5.4. acl(x) = acl(—χ) if and only i/aflM = aflM', where Μ /' = ker(-/i,x).

=

References [AB]

H. Abels and K. S. Brown, Finiteness properties of solvable S-arithmetic groups: An example, J. Pure Appl. Algebra 44 (1987), 77-83.

[Abe]

H. Abels, Finiteness properties of certain arithmetic groups in the function field case, Israel J. Math. 76 (1991), 113-128.

[Abr]

P. Abramenko, Finiteness of properties of Chevalley groups over ¥q[t], Israel J. Math. 87 (1994), 203-223.

[B]

K. S. Brown, Trees, valuations, and the Bieri-Neumann-Strebel invariant, Invent. Math. 90 (1987), 479-504.

[BB]

M. Bestvina and N. Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), 445-470.

22

Robert Bieri

[BG]

R. Bieri, R. Geoghegan, Kernels of actions on non-positively curved spaces, in: Geometry and Cohomology in Group Theory (Kropholler, P. H., Niblo, G. A., Stöhr, R., eds.), London Math. Soc. Lecture Note Ser. 252,24-37, Cambridge Univ. Press, 1998.

[BNS]

R. Bieri, W. D. Neumann, and R. Strebel, A geometric invariant of discrete groups, Invent Math. 90 (1987), 451-477.

[BR]

R. Bieri and B. Renz, Valuations on free resolutions and higher geometric invariants of groups, Comment. Math. Helv. 63 (1988), 464-497.

[BS1]

R. Bieri and R. Strebel, Valuations and finitely presented metabelian groups, Proc. London Math. Soc. (3) 41 (1980), 439^64.

[BS2]

—, Geometric invariants for discrete groups, preprint, Frankfurt/M. 1992.

[Gl]

R. Gehrke, Die höheren geometrischen Invarianten für Gruppen mit Kommutatorrelationen, Dissertation, Universität Frankfurt/M., 1992.

[G2]

—, The higher geometric invariants for groups with sufficient commutativity, Comm. Algebra 119 (1997), 297-317.

[Ml]

H. Meinert, The geometric invariants of direct products of virtually free groups, Comment. Math. Helv. 69 (1994), 39-48.

[M2]

H. Meinert, The homological invariants of metabelian groups of finite Prüfer rank: A proof of the Em-conjecture, Proc. London Math. Soc. (3) 72 (1996), 385-424.

[M3]

—, Actions on 2-complexes and the homotopical invariant Appl. Algebra, 119 (1997), 297-317.

[MMV1]

J. Meier, H. Meinert and L. VanWyk, Higher generation subgroup sets and the -invariants of graph groups, Comment. Math. Helv. 73 (1998), 22—44.

[MMV2]

—, On the

[Rl]

B. Renz, Geometrische Invarianten und Endlichkeitseigenschaften von Gruppen, Dissertation, Universität Frankfurt/M., 1987.

[R2]

—, Geometrie invariants and HNN-extensions, in: Group Theory, Proc. of the Singapore Group Theory Conference, June 1987 (ed. K. N. Cheng and Y. K. Leong), 465-484, Walter de Gruyter, Berlin-New York 1989.

[S]

U. Stuhler, Homological properties of certain arithmetic groups in the function field case, Invent. Math. 57 (1980), 263-282.

2

of a group, J. Pure

-invariants of graph products based on trees, preprint

Convergence groups and configuration spaces B. H. Bowditch

Abstract. We develop some of the basic properties of a convergence group acting on an arbitrary compact hausdorff space from the point of view of the induced action on the space of distinct triples. We also look at some topological properties of the space of distinct n-tuples in a continuum.

1. Introduction In this paper, we give an account of convergence groups in a fairly general context, focusing on aspects for which there seems to be no detailed account in the literature. We thus develop the theory from a slightly different perspective to usual; in particular from the point of view of actions on spaces of triples. The main applications we have in mind are to (word) hyperbolic groups, and more generally to Gromov hyperbolic spaces. In later sections, we give special attention to group actions on continua. The notion of a convergence group was introduced by Gehring and Martin [GeM 1 ]. The idea is to axiomatise the essential dynamical properties of a Kleinian group acting on the ideal sphere of (real) hyperbolic space. The original paper thus refers directly only to actions on topological spheres, though most of the theory would seem to generalise to compact hausdorff spaces (or at least to compact metrisable spaces). The motivation for this generalisation stems from the fact that a (word) hyperbolic group (in the sense of Gromov [Gr]) acting on its boundary satisfies the convergence axioms. Indeed any group acting properly discontinuously on a complete locally compact (Gromov) hyperbolic space induces a convergence action on the ideal boundary of the space. More general accounts of convergence groups in this context can be found in [T2,F1,F2]. There are essentially two equivalent definitions of "convergence group". The original, and most readily used, demands that every sequence of distinct group elements should have a "convergence subsequence" (or what we shall call a "collapsing subsequence") with very simple dynamics. The second definition, which we focus on here, demands that the induced action on the space of distinct triples should be properly discontinuous. This is, in many ways, a more natural formulation. The equivalence of these definitions for group actions on spheres is proved in [GeM2]. It would seem that their argument extends without change to actions on (metrisable) Peano continua.

24

B. H. Bowditch

For the general case (compact hausdorff spaces) we shall need a slightly different approach. The second definition can be restricted to give us what we shall call "uniform convergence actions" — where the action on distinct triples is assumed also to be cocompact. The typical example of such an action is that of a hyperbolic group on its boundary. In fact, it is shown in [Bo6] that these are the only such examples. In some ways, this renders further study of uniform convergence actions superfluous. However certain properties of hyperbolic groups seem to fit most naturally into this dynamical context (see, for example [Bo2, Bo3, Bo5]), so it is appropriate to give, as far as possible, a purely dynamical treatment of some of these in the hope of finding broader applications (for example, to relatively hyperbolic groups). Indeed, introducing geometric considerations often does not seem to help significantly anyway. There are various categories of spaces in which one might be interested. For example, with decreasing generality, we have compacta (compact hausdorff spaces), perfect compacta (compacta with no isolated points), continua (connected compacta) and Peano continua (locally connected continua). Every compactum contains a unique maximal perfect closed subset, which in all interesting cases is non-empty (i.e. for nonelementary groups). This is clearly preserved by any group action. We thus do not loose much by restricting to perfect compacta whenever this is convenient. Continua arise as cases of particular interest (for example, boundaries of one-ended hyperbolic groups). Peano continua are much easier to deal with. Most of the standard arguments concerning actions on spheres would seem to generalise unchanged to this context. The passage to the general case of compacta is sometimes slightly less trivial. One of the main motives for writing this paper was to establish some of the groundwork for a deeper study of convergence actions on continua. These arise as important special cases of limit sets and ideal boundaries. There are a number of conjectures which assert that in certain circumstances, such continua are necessarily Peano continua — for example connected limit sets of geometrically finite Kleinian groups in any dimension (which one might generalise to relative hyperbolic groups) or finitely generated Kleinian groups in dimension 3. (The intersection of these cases, namely 3-dimensional geometrically finite Kleinian groups is already known, see [AnM].) It was also conjectured in [BesM] that the boundary of a one-ended hyperbolic group is locally connected. They showed that this is the case if there is no global cut point. The latter now follows from the results of [Bo2, Bo4, L, Sw]. A more general criterion for the non-existence of global cut points is given in [Bo5], which also has applications to geometrically finite Kleinian groups, and might shed some more light on the first conjecture mentioned above. With these, and other, potential applications in mind, it seems appropriate, as far as possible, to deal with general continua, without any local connectedness assumption. There is another direction in which one might want to restrict the category of spaces under consideration. Most standard arguments (see [GeMl, ΤΙ, Τ2] etc.) make reference to convergence sequences, and thus effectively make some assumption about the order types of neighbourhood bases, or metrisability. Indeed, all likely applications are to metrisable spaces. However some natural constructions (for example those of

Convergence groups and configuration spaces

25

Section 6) cannot be guaranteed to keep us in the metrisable category. For this reason, we shall avoid making any such hypothesis. Since we do not need any "diagonal sequence" arguments, this simply entails replacing the term "sequence" by "net", and "subsequence" by "subnet". The arguments can be translated back into more familiar terms simply by inverting this transformation. As mentioned earlier, the original motivation for the study of convergence groups concerned Kleinian groups, i.e. groups acting properly discontinuously on hyperbolic space (see for example, [Mas] or [Ni]). More generally, one could consider groups acting on manifolds of pinched negative curvature. Many of the definitions concerning types of limit points etc. can be interpreted in the context of convergence groups. For example one can give a definition of geometrical finiteness intrinsic to the action of the group on its limit set (see [Bol], generalising the description given in [BeaM].) Thus, a group is geometrically finite if and only if every limit point is a conical limit point or a "bounded" parabolic fixed point. Such definitions make sense for convergence groups, though it's unclear to what extent the standard results generalise to this case, or indeed to what extent such generalisations would be genuinely useful. However, in certain cases, in particular uniform convergence actions (corresponding to convex cocompact Kleinian groups) such generalisations seem natural, and leads to some powerful techniques for their study. As we have already mentioned, any uniform convergence group acting on a perfect compactum is hyperbolic, and the action is topologically conjugate to the action of the group on its boundary. One can, however, deduce many properties of such actions directly from these dynamical hypotheses, for example, the non-existence of parabolics. If the space is a Peano continuum, then we see that the group must be one-ended. One can go on to derive the JSJ splitting from a (mostly elementary) analysis of the local cut point structure [Bo3]. As mentioned earlier, the fact that any continuum admitting a uniform convergence action is necessarily locally connected requires a lot more work, including knowing the group is hyperbolic. The specific cases of convergence groups acting on spheres have been much studied. In particular, the work of Tukia, Gabai and Casson and Jungreis [Tl], [Ga], and [CasJ] tells us that any convergence group acting on a circle is conjugate to a fuchsian group. (This result relies on the earlier analysis of convergence actions on the 2-disc by Martin and Tukia [MarTl]. It is, in turn, an essential step in the proof of the Seifert conjecture for 3-manifolds — see [Me].) One can similarly ask if every uniform convergence group Γ, acting on the 2-sphere, S2, is conjugate to a cocompact Kleinian group. Some significant progress in this direction has been made by Cannon and coworkers (under the assumption that Γ is hyperbolic). See for example [CanS]. (We remark that it is known if Γ is quasiisometric to hyperbolic 3-space [CanC].) We also note that [Bo6] together with Stallings's theorem on ends [St] and Dunwoody's accessibility theorem, tells us that any uniform convergence group acting on a Cantor set is finitely generated virtually free. We shall give precise definitions of convergence groups in Section 2. In fact, these definitions make sense for any set of homeomorphisms — closure under composition

26

B. H. Bowditch

or inverses is irrelevant in this regard. We should note that we are using the term "convergence group" for what was called a "discrete convergence group" in [GeMl]. The term "discrete" has frequently been omitted in the subsequent literature, and the more general notion of "convergence group" described in the original paper will not concern us here. An outline of the paper is as follows. In Section 2, we prove the equivalence of the two definitions of convergence group in a general setting. We show that a properly discontinuous action of a group on a complete locally compact hyperbolic space extends to a convergence action on the boundary. In Section 3, we give a brief outline of the standard results concerning convergence actions. In Section 4, we consider particular categories of limit points, in particular, conical limit points. In Section 5, we discuss various equivalent formulations of quasiconvexity for subgroups of a uniform convergence groups. In Section 6, we consider how properly discontinuous cocompact actions of a group can be compactified by uniform convergence actions. In Section 7, we consider connectedness properties of configuration spaces in continua. Of particular interest is the space of distinct triples. From this, we can see directly that a group acting as a uniform convergence group on a Peano continuum is finitely generated and one-ended. Most of the work for this paper was prepared while visiting the University of Melbourne as part of the Special Year in geometric group theory. I would like to thank Craig Hodgson and Walter Neumann for the invitation, and the many other members of the geometry group for their hospitality. I would also like to thank Eric Freden for his comments on this paper.

2. Convergence groups and spaces of triples In this section, we give two definitions of the "convergence property" of a set, Φ, of homeomorphisms of a compactum, M, to itself. In the case where Φ is a group, this defines the notion of a "convergence action" or "convergence group", though for most of this section, there will be no need to assume closure under composition or inverses. The equivalence of these definitions is shown in [GeM2] for groups of homeomorphisms of spheres. Their argument would seem to generalise unchanged to the case where Μ is a (metrisable) Peano continuum. For the general case where Μ is any compactum, we shall use a slightly different argument. We begin by introducing the space of distinct triples. Let Μ be (for the moment) any hausdorff topological space. We give the space of ordered triples, M3, the product topology. Let Δ c Μ3 be the large diagonal, i.e. the (closed) subset of triples have at least two entries equal. Let Θ°(Μ) = Μ3 \ Δ be the space of "distinct ordered triples". There is a natural continuous surjective map Δ —> M which sends any triple with at least two entries equal to χ to the point χ G M. We denote the quotient by

Convergence groups and configuration spaces

27

3Θ°(Μ). Thus, 9Θ°(Μ) may be naturally identified with M. In fact, we can define an equivalence relation on M3 by deeming two triples to be equivalent if two entries of the first triple are both equal to two entries of the second triple. Clearly, this relation is trivial (i.e. equality) on Θ°(Μ). We may thus identify the quotient space as a union Θ°(Λ/) U 9Θ°(Μ). This quotient is hausdorff, and contains 9Θ°(Μ) as a closed subset. If M is perfect, then Θ°(Μ) is dense in Θ°(Λ/) U 9Θ°(Μ). If M is compact, then so is Θ°(Λί) U 3Θ°(Μ). Also, if M is locally compact, then so is Θ°(Μ). Now, the symmetric group on three letters acts on Λ/3 by permuting the coordinates. This induces an action on Θ°(Μ) U 9Θ°(Μ), which is trivial on 9Θ°(Μ). We write the quotient as Θ(Μ) U 3Θ(Λί), where Θ(Μ) is the quotient of Θ°(Μ). Again, 9Θ(Μ) is closed, and may be naturally identified as M. We think of an element of Θ(Μ) as a "distinct (unordered) triple", i.e. a subset of M of cardinality 3. Suppose now that Μ, Ν are compacta (compact hausdorff topological spaces), and that Φ is a set of homeomorphisms of M onto N. (Of course, we could take M = N, but we don't want to distinguish any preferred identity homeomorphism.) We write Φ -1 = {φ~] | φ e Φ}. Note that each φ e Φ induces a homeomorphism of Θ(Μ) onto Θ(ΛΟ, which we shall also denote by φ. In actual fact, we don't really want to assume that all the elements of Φ are distinct homeomorphisms (since we shall eventually want to allow for group actions with non-trivial kernel). To be more formal we should really view Φ as a collection of homeomorphisms with some indexing set, though to do so explicitly would only confuse our notation. In what follows we shall assume that M (and hence N) has at least 3 points. Definition 1. We say that Φ is properly discontinuous on triples if, for all compact subsets Κ c 0(M) and L c &(N), the set {φ € Φ | φΚ Π L φ 0} is finite. Definition 2. If Φ' c Φ, α e M and b e N, we say that Φ' is a collapsing set with respect to the pair (a, b) if, for all compact subsets Κ C. M \{a} and L C N \ {b}, the set (φ e Φ' \ φΚ Π L ^ 0} is finite. We say that Φ' is a collapsing set if it is a collapsing set with respect to some pair (a, b). Note that the pair (a, b) for a given collapsing set is uniquely determined. We shall refer to a and b, respectively, as the repelling and attracting points of the set Φ. (This terminology becomes more natural, when we reformulate this in terms of nets.) Note also that if Φ' is a collapsing set with respect to a pair (a, b), then (Φ')-1 is a collapsing set with respect to the pair (b, a). Definition 3. We say that Φ has the convergence property if every infinite subset Φ' C φ contains a further infinite subset Φ" c φ' which is a collapsing set. We note that if Φ has the convergence property, then so does Φ"1, as well as any infinite subset of Φ. This statement is also true of the property of being properly discontinuous on triples. Our first objective will be to show that these notions are equivalent:

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Proposition 2.1. An infinite set of homeomorphisms of a compactwn has the convergence property if and only if it is properly discontinuous on triples. The "only if" part is elementary and well-known. The "if" part is also known at least for spheres (and metrisable Peano continua). The general case involves a bit more work. Before we give the proof we shall rephrase the definitions in a form that is more convenient to work with. The convergence property is usually phrased in terms of sequences. For the general case, we shall use nets. We begin by recalling a few standard (and not so standard) definitions concerning nets and subnets. Let Z be any set. A net in Z is a map, [n H* zn], from a directed set, (D, no} for some «o £ D. We say that a property is true/or all sufficiently large n if it is true for all n in some final segment. A subset of D is cofinal if it meets every final segment. We say that a net, (zn)n, is wandering if for all z € Ζ, ζη Φ ζ for all sufficiently large n. If (/, / and φηζη ->· ζ'. If χ, y, ζ are all distinct, then x', yf, zf cannot all be distinct (and so also conversely). To reformulate the convergence property, we proceed as follows. We say that a net, (φϊ)ϊ, of elements of Φ is a collapsing net if there are points α € Μ and b € N such that the net of maps φι\Μ \ {a} converges locally uniformly to the point b. We shall denote this by φι\Μ \ {a} -» b. Note that a collapsing net is necessarily wandering. Local uniform convergence can, in turn, be expressed in terms of nets. Thus, φι·\Μ \ {a} does not converge locally uniformly to b if and only if there is some

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subnet (0y );- of (0, ) , , and a net (xj )y of points of M such that Xj —> χ and φj Xj —> x', where χ e Μ \ {a} and x' e Ν \ {b}. We see that Φ has the convergence property if and only if every wandering net of elements of Φ has a collapsing subnet. In what follows, we shall freely pass to subnets without necessarily changing notation. We justify such liberties with phrases such as "without loss of generality". We now set about proving Proposition 2.1. One direction is easy: Lemma 2.2. If Φ has the convergence property, then it is properly discontinuous on triples. Proof. Suppose (φη)η is a wandering net of elements of Φ. Suppose we can find nets (xn)n, (yn)n and (z„)„, of elements of M such that xn -> x, yn -^ y, zn ->· z, φηχη -*· x', Φnyn -> / and φηζη -+ ζ' , where je, y, z e M are all distinct, and x',y',z'eN. After passing to a collapsing subnet, we can find points α e M and b € N such that φη \M \ {a} -» b. Moreover, we can assume that x , y φα. It follows that φηχη —> b and · {a, b} to mean that for all neighbourhoods Ο 3 a and U 3 b, we have un e O U U for all sufficiently large n. Let's now suppose that Φ is properly discontinuous on triples. In the following lemmas, (x„)„, (v„)„, (z„)„ and (w„)„ are assumed to be nets in M, and (φη)η is a wandering net in Φ. Lemma 2.3. Suppose xn —> x, yn —> y, zn —> z, withx,y,z distinct. Suppose φηχη -> x' andφnyn -> y' with χ' φ y'. Then φηζη -> {x', y'}· Proof. Otherwise some subnet οίφηζη would converge to a point z' £ {x', y'}.

Π

Lemma 2.4. Suppose xn —> x, yn —> y, zn —>· ζ, ννά/ι *,_>>, z distinct, and that φηχη -* a, 0„v„ -> α αηάφηζη -+ b φ a. Ifw„ -> M; / z, /Ae/i 0„iy„ -> {a, fc}. Proof. Without loss of generality, w φ y, so we can apply Lemma 1.3 (replacing xn b y z n , a n d z n by w„). D Lemma 2.5. Suppose xn —>· x, yn —> y, z« -> ζ α«ί/ w« —>· w, wiVA x,y, z,w all distinct. Suppose φηχη —> α, φηγη —> α, φηζη —> b αηάφηυ]η —>· b. Then a = b. Proof. Choose any c €. N \{a, b}, and let un =φ~{c. Passing to a subnet, (u„)n can be assumed to converge to some point u e M. Now, either u φ {χ, ν} or M g {z, w}. If α φ b, then applying Lemma 2.4, we derive, either way, the contradiction that c = φnun ->· {a, 6}. D Lemma 2.6. Suppose x,y,z e M are distinct, and zn —> z. Suppose that φηχ —> a, 0 n y —>· a ana1 φηζη ^ b ^ a. Then φη\Μ \ { z ] converges locally uniformly to a.

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Proof. First, we prove pointwise convergence. Let w € M \ {z}. By Lemma 2.4, we have φηνύ —> {α, &}. If φηνυ -ft- α, then, passing to a subnet, we can suppose that φη w —> b, contradicting Lemma 2.5. To prove locally uniform convergence, suppose (maybe after passing to a subnet) that wn —»· u; ^ z. By pointwise convergence, we can suppose that u; ^ {x, y } . (Otherwise replace χ or y by some other point of Μ \ {z}.) Again by Lemma 2.4, we have φη^η —> {«, b}. If u;„ /*· a, we get a contradiction to Lemma 2.5 as before. D Lemma 2.7. Φ has the convergence property. Proof. Let (φη)η be any wandering net in Φ. We want to find a collapsing subnet. Choose any triple *, y, z of distinct elements of M. Passing to a subnet, and permuting x, y, z if necessary, we can assume that φηχ —> α,φηγ —> a and φηζη —>· b for some a, b e N. If α φ b, then Lemma 2.6 tells us immediately that φη\Μ \{z} -» a. We can thus assume that a = b. Choose any point c e N\{a}, and let wn = φ~l c. Passing to a further subnet, we can suppose that wn -> w e M. Without loss of generality, w £ {x, y } . In this case, Lemma 2.6 tells us that φη\Μ \ {u;} -»a. D Lemmas 2.2 and 2.7 together prove Proposition 2.1. In fact, we can strengthen Lemma 2.7 as follows: Proposition 2.8. If Φ is properly discontinuous on triples, then Φ has the convergence property on Θ(Μ) U d Θ (Μ). Proof. We know, by Lemma 2.7, that any wandering net in Φ has a collapsing subnet in Μ = 9Θ(Λ/). We are thus reduced to considering a net, (φη)η, in Φ such that 0„|30(M) \ {a} -» /?, for some a e 30(M) and b e 3θ(ΛΓ). We claim that 0„|(0(M) U 3Θ(Μ)) \ {α} -» fr. Suppose that (,·),· is any subnet, and (#/),· is a net in Θ(Μ) U 3Θ(Μ), which converges to some θ e Θ(Μ) U 3Θ(Μ) \ {α}. We claim that 0,0, converges to b. We can partition the domain of the net into two subdomains depending on whether θί lies in Θ(Μ) or 9Θ(Μ). This gives us (at most) two subnets, and it's enough to verify the claim for each of these. (Of course there's no reason to suppose that both subsets of the domain are cofinal, but if one isn't then there's nothing to verify in that case.) In fact, we know by construction that the claim is true for the subnet lying in 30(M), so we can assume, without loss of generality that #, e Θ(Λ/) for all i. We write 6»; = { x i , y i , Z i } . Suppose first, that θ € Θ(Μ). Write θ = {x, y, z}. We can assume that χ, -> x, y, ->· y and Zi -> z (since we are free to label the enties in the triple #, as we choose). Also without loss of generality, x, y φ a. It follows that ,*,· —> b and /y,· —> b, and so 0/0,· -^ bin Θ(Μ) U 3 Θ (Μ) as claimed. We can thus assume that θ € 3Θ(Μ) \ {a}, so that θ corresponds to some point x € Μ \ {α}. We can assume that Λ:, —> x and y, —>· x. Since χ φ α, we have φίΧί —> a and 0,y, -> am M. Thus, again 0;0/ —>· as claimed. D

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This concludes the basic observations about sets of homeomorphisms. The cases of interest here concern group actions. Suppose that M is a compactum, and that Γ is a group acting by homeomorphism on M. Definition 4. We say that Γ is a convergence group (or that the action is a convergence action) if, as a set of homeomorphisms, it has the convergence property. We see that Γ is a convergence group if and only if the induced action on Θ(Μ) is properly discontinuous. Moreover, this implies that the induced action on Θ(Μ) U 9 Θ (Μ) is also a convergence action. There are two subtleties we should remark upon. The first is that we have not assumed that Γ acts effectively. Thus, we should more formally view the set of homeomorphisms in the above definitions as a collection indexed by Γ. In any case the definitions imply that the action should have finite kernel, so the distinction is not really important. The second point is that we have assumed that Μ has at least 3 elements. The appropriate definition of a convergence action on a smaller set may be open to debate, but it would seem natural to allow any action on a singleton, and any virtually cyclic action on a pair. The following is a trivial, but useful observation: Lemma 2.9. Suppose the group Γ acts by homeomorphism on locally compact hausdorff spaces X and Y. Suppose f : Υ —> X is a proper surjective Γ -equivariant map. Then Γ acts properly discontinuously on X if and only if it acts properly discontinuously on Y. Also, Γ acts cocompactly on X if and only if it acts cocompactly onY. We finish this section with two applications of this. The first concerns quotient spaces. Suppose Μ, Ν are compacta, and /: M —> Ν is surjective. Let Θ/ν(Λί) C Θ(Μ) be the subset of triples { x , y , z } such that f x , f y , f z are all distinct. We see that / induces a natural surjective map, Θ/: Θ^(Μ) —> Θ (W), given by Suppose now that Γ acts on M and N, and that / is Γ-equivariant. It follows that Θ/ν(Μ) is a Γ-invariant subset of Θ(Μ), and that Θ/ is Γ-equivariant. If Γ acts properly discontinuously on Θ(Λ/), then it does so on ©^(M), and hence, by Lemma 2.9, on 0(JV). We deduce: Proposition 2.10. Suppose that Γ acts on the compacta Μ and N, and that f : M —>· Ν is a Γ-equivariant map. If Γ acts as a convergence group on M, then it acts as a convergence group on N . One can can give an alternative (perhaps simpler) proof of this result using the collapsing subsequence definition instead. (This is set out explicitly in [Bo2].) The second application involves induced action on boundaries of hyperbolic spaces, as defined by Gromov [Gr]. For the necessary background, see, for example, [GhH].

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Suppose that (X, d) is a complete, locally compact path-metric space which is (Gromov) hyperbolic. Thus, any closed metric ball in X is compact. Also, X can be compactified in a natural way by adjoining its (Gromov) boundary, dX. Thus, X U dX carries a natural compact topology. It is also metrisable, though does not admit any preferred metric. Suppose a group Γ acts properly discontinuously and isometrically on X. We get an induced action by homeomorphism on X U dX. We claim that this is a convergence action. For the purposes of future reference, we split this into two parts. Lemma 2.11. Γ acts as a convergence group on dX. Proof. Let k be the constant of hyperbolicity of X (in the sense that for any geodesic triangle in X, there is a point a distance at most k from each of its edges). Let Υ c χ χ &(dX) be the subset of pairs, (a, {x\, Χ2, ^3}) such that there exist biinfinite geodesies, «i,Q!2, £*3, with or, connecting ;c( tojc i+ i (with subscripts mod 3) such that d (a, or,·) < k for each i e {1,2,3}. Thus, Υ is a closed subset of Χ χ Θ (3X). Moreover, the natural projections of Υ to X and to Θ(9Χ) are both proper and surjective. Also the whole construction is natural and hence Γ-equivariant. Since Γ acts properly discontinuously on X, it follows, by Lemma 2.9, that it does so also on Υ and hence on Θ(9Χ). D Proposition 2.12. Γ acts as a convergence group on X U dX. Proof. (Since X U dX is metrisable, we may as well phrase everything in terms of sequences.) Suppose that (γη~)η is a sequence of distinct elements of Γ. Since Γ acts as a convergence group on dX (Lemma 2.11), we can find a subsequence, (χ,·),· and α, b e dX, such that χ,· |3X \ {a} -» b. We claim that γι·\(Χ U dX) \ {a} -» b. To see this, suppose that K c (X U 3Χ) \ {α} is compact. Suppose that (jt;)/eN is any sequence in K. We can find a compact subset L c dX \ {a} and points v,·, zi e L such that each *,· lies in the (closed) biinfinite geodesic joining y, to z,. Now y, —>· £ and z,· —* £, and so it follows easily that x, —>· b as required. D This result can be compared with Proposition 6.6, where Γ is assumed to act cocompactly, but X is not assumed to be metrisable. We finish this section by introducing "uniform convergence groups" to which we shall return again later. Suppose that Γ acts by homeomorphism on a perfect compactum, Μ. Definition 5. We say that P i s a uniform convergence group if it acts properly discontinuously and cocompactly on the space of distinct triples, &(M). We refer to the action as a "uniform convergence action". The typical examples arise as boundaries of hyperbolic groups, as we shall see below. Suppose that Γ is a (word) hyperbolic group. Its boundary, 3Γ, is a compact metrisable space, on which Γ acts by homeomorphism. Indeed, if (X, d) is any complete locally compact hyperbolic space, and Γ acts properly discontinuously and

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cocompactly on X, then we may naturally identify the (Gromov) boundary, dX with 3Γ. The typical example of such an X is the Cayley graph of the given hyperbolic group with respect to any finite generating set. (One could also take the Rips complex etc., see [GhH, BesM].) We already know (Lemma 2.11) that Γ acts as a convergence group on 3Γ. This is also shown in and [Fl] and [T2]. In this case our previous argument gives us the additional information: Proposition 2.13. A hyperbolic group Γ acts as a uniform convergence group on its boundary, 3Γ. Proof. Let A" be a Cayley graph of Γ, so that we can equivariantly identify 3 Γ and dX. As in the proof of Lemma 2.11, we construct a locally compact hausdorff space, Κ, and proper equivariant surjections of Υ to X and to Θ(3Χ). Since the action on X is properly discontinuous and cocompact, we see, by Lemma 2.9, the same is true of the action on Θ ( 3 X ) . D This result well known (though I've not found an explicit reference). The converse was, for a time, an open problem, though it appears that Gromov had long been confident that this was indeed true. A proof is given in [Bo6].

3. General properties of convergence groups In this section, we briefly outline how one may develop the theory of convergence groups on general compacta. Most of the results stated here are well known, and accounts can be found in [GeMl] and [T2] (see also [Tl], [Fl] and [F2]). Some of these are given in slightly restricted contexts, though the arguments would seem to generalise unchanged. The proofs are typically based on the "collapsing net" (or "convergence subsequence") definition of convergence group. Typically, the development proceeds via a classification of elements according to their dynamics, a discussion of "elementary" subgroups, the partition of the space into limit set and discontinuity domain etc. Here, we shall only concern ourselves with aspects relevant to the rest of this paper. Suppose that Γ acts as a convergence group on the compactum, M, with card Μ > 3. Given γ e Γ, we write fix γ = {χ € Μ \ γ χ = χ}. Definition 6. We say that an element of Γ is elliptic if it has finite order. We say that γ e Γ is parabolic if it has infinite order, and card fix γ — 1. We say that χ e Γ is loxodromic if it has infinite order, and card fix γ = 2. Clearly these possibilities are mutually exclusive. The following is the most basic result about convergence groups. The proof we give here is more or less copied from [T2].

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Lemma 3.1. Every element of Γ is elliptic, parabolic or loxodromic. Proof. Suppose γ e Γ has infinite order. Since (γ) acts properly discontinuously on distinct triples, we see that card fix χ < 2. It thus suffices to show that fix γ φ 0. Consider the sequence of elements (y n ) n e N· There is a subnet, (y'),·, and points a,b Ε Μ such that y'|M \ {a} -» b. Choose any c e Μ \ {α, y"1«}· Now, y'c —> b, so y i + 1 c = yy'c -> yb. But, y' + 1c = y ! y c "*· ^» since yc 7^ a. Thus, yfc = 6. D One can go on to show that, in fact, (y")« £ N is itself a collapsing sequence. In particular, {y} acts properly discontinuously on M \ fix y. If y is parabolic with fixed point p, we see that, for all Λ € Μ, γηχ -> ρ as η ->· οο and as η -» -σο. If y is loxodromic, we can write fix y = {fix+ y, fix" y}, such that y"|M \ {fix~ y} -» fix+ y. It's not hard to see that, in this case, (y) acts cocompactly on Μ \ fix y. Note that every power of a parabolic is parabolic, and every power of a loxodromic is loxodromic. It's known that a loxodromic cannot share a fixed point with a parabolic. Also if two loxodromics share a fixed point, then they have both fixed points in common. Moreover the setwise stabiliser of any pair of points is virtually cyclic. (For proofs, see for example [T2].) We may summarise these results as follows: Lemma 3.2. Suppose an infinite subgroup, G < Γ fixes some point ρ e M. Then, G either consists entirely elliptic s and parabolic s, or consists entirely ofelliptics and loxodromics. In the latter case G also fixes some other point, q € M \ {p}, and is virtually cyclic. These cases are mutually exclusive. We refer to them respectively as "parabolic" and "loxodromic". In fact, in the parabolic case, G acts properly discontinuously on M \ {p}. In the loxodromic case, it acts properly discontinuously and cocompactly on M \ { p , q } . We shall refer to a subgroup G < Γ, as elementary if it is finite, or preserves setwise a nonempty subset of Μ with at most 2 elements. It is shown in [T2] that every non-elementary subgroup contains a free subgroup of rank 2. It is conceivable in the "parabolic" case of Lemma 3.2, that G may contain only elliptic elements. There are no other possibilities for infinite torsion subgroups of Γ. We remark that, from the result of [D2], any finitely generated inaccessible group must contain an infinite torsion subgroup. If we can rule out such possibilities (for example, for convergence actions on the 2-sphere), we can deduce that Γ is accessible. Note that any element which commutes with a loxodromic must preserve setwise its fixed point set. From this, it's a fairly easy deduction that: Proposition 3.3. Any infinite virtually abelian subgroup of Γ has a subgroup of index at most 2 which fixes a point. The following is shown in [T2]:

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Lemma 3.4. Suppose that U C M is open, with closure, . Suppose γ Ε Γ with γ c U. Then γ is loxodromic (with fix+ γ E U and fix" γ E M \ ). The idea of the proof is to note that the sets f}™=0 Yn V and f|^o Y~"(M\U) are non-empty disjoint closed y-invariant subsets. It's then easy to see that they must both be singletons, and hence fixed points of γ. In particular, we deduce: Lemma 3.5. If (γη) η is a net in Γ with γη\Μ \ {a} -» b, where α φ b. Then, γη is loxodromic for all sufficiently large n. Proof. Let U be an open neighbourhood of b with a £U. For all sufficiently large n, we have γηϋ c (7. For such η, γη is loxodromic. D The next natural step in working with convergence groups is to define a natural partition of M into a limit set, Λ, and discontinuity domain, Ω = Μ \ Λ. The limit set can be defined as the set of limit points, where a limit point is an accumulation point of a Γ-orbit. In other words, χ Ε Λ if and only if there is a net, (y„)„, in Γ, and a point y Ε Μ \{x}, such that yny —> x. Thus, Λ is closed, and Ω is open. If Γ is infinite, then Λ is non-empty. If we assume that Γ is non-elementary, then Λ is perfect and Γ acts minimally on Λ. In fact, Λ is the unique minimal non-empty closed Γ-invariant subset of M. In contrast, Γ acts properly discontinuously on Ω. (Many actions we will be considering will be non-elementary and minimal, i.e. Ω = 0. Note that this implies that Μ is perfect.) In the next section we shall be considering particular classes of limit points.

4. Conical limit points In studying Kleinian groups, it has proved important to distinguish different classes of limit points. These are discussed, for example, in [Mas] and [Ni]. Obvious examples of such classes are parabolic and loxodromic fixed points. A particularly important class (including the loxodromic fixed points) are "conical limit points" (also known as "radial limit points" or "points of approximation"). For example, they arise naturally in the study of conformal densities (as discussed in [Ni]). Also, one can characterise the property of geometrical finiteness dynamically, by demanding that every limit point is a conical limit point or a "bounded" parabolic fixed point (see [BeaM, Bol]). In the context of convergence groups acting on spheres, conical limit points have appeared in [MaiTl] and [MarT2]. In this section, we define a natural notion of conical limit point for convergence groups, which reduces to the standard notion in the case of Kleinian groups. It will be immediate from the definition that, in the case of a uniform convergence action, every point is a conical limit point. For Kleinian groups, the converse also holds.

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(This amounts to the statement that a geometrically finite group with no parabolics and empty discontinuity domain is cocompact.) I don't know if this converse is true in general. Suppose, then, that Γ acts as a non-elementary convergence group on a compactum, M. Definition 7. A point χ € Μ is a conical limit point if there are nets (xn}n and (γη)η in Μ \ {jc, y} and Γ respectively with xn —> χ such that there exists y e M\{x} such that γη(χ, y, xn) remains in a compact subset of Θ°(Μ). (In fact, as we shall see, we get an equivalent definition if we replace the phrase "there exists y € Μ \ {*}" by "for all y e Μ \ {*}".) Note that we can assume that γη(χ, y, xn) converges on some point, (a, b, c) e Θ°(Μ), and that (γη)η is a collapsing net. In fact, we must have γη\Μ \ {x} -» b. (To see this, suppose γη\Μ \ {x1} -» b', so that γ~ι\Μ \ {b'} -» x'. Now (γηχη)η and (γηχ)η converge on different points, at least one of which must lie in Μ \ {b'}, whereas, their images under γ~ι both converge on x. It follows that x = x'. Now, yny —> b, but γ~ι (yny) /» x, so we must have b = b'.) Note that if ζ € Μ \ {χ}, then γη(χ, z, Xn) —>· (a, b, c). This justifies our earlier remark about quantifiers. Note also that x is an accumulation point of some Γ-orbit. We see: Proposition 4.1. A conical limit point is a limit point. We should note that the property of being a conical limit point is intrinsic to the action of Γ on the limit set, Λ. (That is, a point x e Λ is conical limit point for the action of Γ on Μ if and only of it is a conical limit point for the action of Γ restricted to Λ.) This is easy to see, noting that Θ°(Λ) is a closed subset of Θ°(Μ). In [Bol], we gave another, equivalent definition of conical limit point. Namely, we said that x € Λ is a conical limit point if and only if there is a wandering net, (γη }n in Γ, such that for all y e Λ \ {jc}, the ordered pairs (γηχ, yny) lie in a compact subset of the space of distinct pairs of Λ (i.e. Λ χ Λ minus the diagonal). We can assume that (γη)η is a collapsing net, and that (γηχ, yny) converges on some pair (a, b) with α φ b. Now, either γη |Λ \ {x} -» b or γη |Λ \ {y} -» a. However, the latter cannot occur, since choosing any ζ e Λ \ {x, y } , we would get (γηχ, γηζ) -> (a, a). If we now fix any c e Λ \ {a, b}, and let xn = γη [c, we see that xn —> x. We thus arrive at our original definition of a conical limit point. The converse statement is elementary. The following is a standard result in the case of Kleinian groups. (A proof for 3-dimensionial Kleinian groups is given in [BeaM] or [Mas], and generalised to any dimension in [SuS].) I'm indebted to Pekka Tukia for suggesting a means of significantly simplifying my original argument. Proposition 4.2. A conical limit point cannot be a parabolic fixed point. Proof. Suppose, to the contrary, that ρ e Μ is both. Thus, there is a parabolic, β e Γ, with fixed point p. Moreover, there is a point, q e M, and nets (xn)n and (γη)η in Μ\{jc} and Γ respectively, withjc„ -» pandwithy„(p, q, xn} -> (a,b,c) e Θ°(Μ).

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As discussed above, we have γη\Μ \ {p} -» b. We can suppose that γηρ φ b for alln. Now, fix for the moment some n, and consider the net (γ~ι yn)m (where m ranges over the same directed set). Now, γ~' |Μ \ {b} -» p, and so γ~' γη \Μ \ {γ~lb} —>· p. Since y„~lb φ ρ, we see, by Lemma 3.5, that γ^ιγη is loxodromic for all sufficiently large m. Now, let 8n = Yn Y n ^ · We claim that the net (8n)n is wandering. For suppose not. This means that there is some η such that the set of m > n with 8m = 8n is cofinal. From the last paragraph, we can find some m > n with 8m = 8n and with Ym^Yn loxodromic. Now, β commutes with γ^Υη- But by Lemma 3.2, a parabolic cannot commute with a loxodromic. This contradiction shows that (δη)η is wandering as claimed. We can thus assume that (8n )„ is a collapsing net. Now, γηρ -* a and δη(γηρ) = γηρ -»· a. Also ynq -»· b and 8n(ynq) = Yn( q) —> £· We see that either })) is quasiconvex in the geometric sense. Moreover the constant of quasiconvexity is a function only of the hyperbolicity constant. More generally, suppose that A" c 3Γ is closed. Now, 0ar(^0 = (J{®9r({*> | x, y e Κ, χ φ y } . Thus, V(0 9r (/O) is a union of sets of the form V(&dr({x, which we showed, in the last paragraph, to be uniformly quasiconvex. Now, x is an ideal point of V(&ar({x, y})). It follows that for any two sets in this collection there is a third which shares an ideal point with each. From this, it's a simple geometric argument to see that their union is (geometrically) quasiconvex. We have shown: Lemma 5.2. If Κ c 3Γ is closed, then V(©ar(^0) is (geometrically) quasiconvex. A subgroup, G, of Γ is geometrically quasiconvex if the G -orbit of some (and hence every) point of V is quasiconvex, or equivalently if there is a G-invariant quasiconvex subset, β c V, with Q/G finite. We can now prove the equivalence of this with our dynamically defined notion. Lemma 5.3. Suppose Γ is hyperbolic, and G < Γ. Then G is geometrically quasiconvex if and only if it is quasiconvex (by our earlier definition) with respect to the action ofFondr. Proof. First note that Γ contains no infinite torsion subgroup. Thus, every elementary subgroup of Γ is quasiconvex by either definition, so we can suppose that G in nonelementary.

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Suppose, first, that Λ c θ Γ is a non-empty closed G-invariant subset with 0 9 r (A)/G compact. Thus, V(0 9 r (A))/G is finite. By Lemma 5.2, V(0 9 r (A)) is quasiconvex. It follows that G is geometrically quasiconvex. Conversely, suppose that G is geometrically quasiconvex. Let Q C V be a Ginvariant quasiconvex subset with Q/G finite. Note that the limit set, AG, is precisely the set of ideal points of Q. Choose any a e V. Now it's easily seen that for any r > 0, the set of γ € Γ such that d(y~la, Q) = d(a, γ Q) < r lie in finitely many left cosets of G in Γ. >From this its easy to see that the collection of Γ-images of AG, indexed by the left cosets of G, is discrete on distinct pairs. Thus, G is quasiconvex by our original definition. D We noted earlier that a quasiconvex subgroup acts as a uniform convergence group on its limit set. This suggest an alternative definition. One might define a subgroup, G, of Γ to be quasiconvex if it is elementary, or if there is a nonempty closed perfect G-invariant subset, A c 8Γ, such that 0(A)/G is compact. Again, A is necessarily the limit set of G, so this is the same as asserting that G acts as a uniform convergence group on its limit set. (We don't really need to assume that A is perfect in the definition, since any closed subset has a natural perfect closed subset, which is non-empty in the case where G is non-elementary.) To show that this apparently weaker definition is equivalent to the standard one, suppose G is non-elementary, and that (9(A)/G is compact. Let A = V(0(A)) and Β = V(0 9 r(A)). We know that A/G is finite. If we can show that B lies inside a uniform neighbourhood of A, then it follows that B/G is finite, and so 0 9 r(A)/G is compact, as required. Suppose, to the contrary, that there is a sequence (£,·)/ 6 N> of points of Β with d(bi, A) —>· oo. Let a, be the nearest point of A to &,·. Since A/G is finite, we can suppose, after translating by elements of G, and passing to a subsequence, that a, = a is constant. Moreover, we can suppose that (6,·),· converges on some point b e 9Γ. Now, each bj lies a bounded distance from a geodesic connecting a pair of points of A. A simple geometric argument shows that by choosing one element from each such pair, we can find a sequence of points of A tending to b. This shows that b € A. Now, since A is perfect, we can find a sequence, (*,),·, of points of Λ \ {b} tending to b. Fix any point, y e A \ {b}, and let c/ be a centre for the three points b, y, jc,. Thus C( e A, and c, —>· b. Moreover the points c\ all lie a bounded distance from a fixed geodesic (namely one connecting y ίο b). Since b, —>· b, a simple geometric argument shows that we can find i, j such that d(bj, a) > d(bj, c y ), contradicting the fact that a is the nearest point of A to b,. This gives the result. I don't know of a purely dynamical proof of this in general. It's not hard to find such a proof in the case where A is connected, using the first definition we gave of quasiconvexity.

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6. Compactifications In this section, we describe how uniform convergence actions naturally "compactify" properly discontinuous cocompact actions. The typical example is that of a hyperbolic group acting on its Cay ley graph, X. We can compactify X as X U 3 Γ, and extend the action of Γ to a convergence action on this space. Moreover the induced action on 3 Γ is a uniform convergence action. Suppose that Μ is a perfect compactum admitting a uniform convergence action by some group Γ. We know from [Bo6] that Γ is hyperbolic, and that Μ is equivariantly homeomorphic to 3Γ. (In [Bo6] we assume Μ to be metrisable. However, the arguments go through without this assumption, replacing sequences by nets. One can therefore conclude, in retrospect, that Μ must be metrisable.) Lemma 6.1. If a group Γ acts as a uniform convergence group on perfect compacta, Μ and N, then there is a unique Γ-equivariant homeomorphism from Μ onto N. Proof. The existence of such a homeomorphism follows from [Bo6], as mentioned above. The uniqueness will be a corollary of Proposition 6.5, though one can give a direct argument as follows. Suppose that g and h are two such homeomorphisms, and that χ e M. Now χ is a conical limit point, so it is the attracting point, in M, of a collapsing net, (γη)η in Γ. Thus, (γη)η = (g ο γη ο g~l)n = (h ο γη ο h~l)n is also a collapsing net for N with attracting point g(x) = h(x}. D This is the only point that we need to make any reference to the the result of [Bo6]. This can be avoided simply by taking the existence of such a homeomorphism as hypothesis where necessary. We give a few definitions. Definition 9. By a compactified space, (X, 3X), we mean a compactum, X U dX, with a partition into two disjoint subsets, X and dX, with X open and dense in X U dX (so that dX is closed). Note that if (X, 3X) is a compactified space, then X is locally compact hausdorff, and dX is a compactum. Definition 10. A morphism f : ( Y , d Y ) —> (Χ, 3X) between two compactified spaces consists of a continuous surjective map, /: Υ U 3 K —> X U 3 Χ, such that f ( Y ) = X and f\dY —> dX is a homeomorphism. Note that f\Y: Υ —> X is a proper continuous surjection. Note also that the composition of morphisms is a morphism. This definition also gives a notion of isomorphism of compactified spaces, where the morphism is assumed to be invertible. Suppose that (Χ, 3Χ) is a compactified space, and that the group Γ acts by homeomorphism on X U 3X, respecting the partition into X and 3X. In other words, Γ acts by isomorphism on the space (Χ, 3Χ).

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Definition 11. We say that Γ acts properly on (Χ, 9X) if the action on X U dX is a convergence action, the action on 3 X is a uniform convergence action, and the action on X is properly discontinuous and cocompact. We see easily that dX is precisely the limit set of the action on X U dX. (In particular, the attracting and repelling point of any collapsing net lie in 9X.) Typical examples of such actions are that of a hyperbolic group, Γ, on (Χ, 9Γ), where X is the Cay ley graph of the group; or that of a group on the space (Θ(Μ),9Θ(Λ/))ίη induced by a uniform convergence action on M. We note: Lemma 6.2. Suppose that Γ acts by isomorphism on compactified spaces (Χ, 9Χ) and (Y, dY), and that f : (Y, dY) —> (X, dX) is α Γ-equivariant morphism. Then Γ acts properly on (X, 9X) if and only if it acts properly on (Y, dY). Proof. This result follows from Lemma 2.9 and Proposition 2.10, except for one point, namely that if Γ acts as a convergence group on X U dX then it acts as a convergence group on Y U dY. To see this, suppose that (γη)η is a wandering net in Γ. Passing to a subnet, we can suppose that γη \(X U 3X) \ {a} -» b, where a, b e 3X. If a 7 , b' e dY are the preimages of a, b e 3X, the we see easily that γη\(Υ U dY) \ {a'} -» b'. D We want to consider the uniqueness of compactifications of properly discontinuous cocompact actions. A useful observation is the following: Lemma 6.3. Suppose the group, Γ, acts properly on compactified spaces, (Χ, 3Χ) and (X', 9X')· Suppose that K C X and K' C X' are compact subsets, and that h: 3X —> dX' is α Γ-equivariant homeomorphism. Then, |J Γ(Κ χ "')Ugraph(/i) is a closed subset of(X U 3X) χ (X' U 3X'). Proof. Certainly, \J V(K χ Κ') and graph(/z) are closed in Χ χ X' and 3X χ 3Χ' respectively. If the conclusion fails, we can find some point, (x, y), of ((X U 3X) χ (X' U 3X') \ (Χ χ X) which lies in the closure of U Γ(Κ χ Κ') and with y φ h(x). Without loss of generality, we can suppose that χ e 3X (otherwise replace h by h~l). We can find nets, (xn)n, Cv«)/? and (γη)η m K, K' and Γ respectively, such that γηχη —»· Λ: and ynyn —>· y. Passing to a subnet, we can suppose that (γη)η is a collapsing net for both X U 3X and X' U 9X', with attracting points a e 3X and b € 9X' respectively. Now, considering the action of Γ on 3X', we see that h(a) is the attracting point of the collapsing net (h o (y„|3X) o h~l)n = (y„|3X')«. We see that b = h(a). But now, returning to X U 9X and X' U 9X', we know that γπ \ Κ and γη \ K' converge uniformly to a and b = h(a) respectively. Thus, γηχη —>· a and ynyn —> b, so χ = a and y = h(a). This gives us the contradiction that y = h(x). D Corollary 6.4. Suppose the group Γ acts properly on the compactified spaces (X, 9 X) and (X', 9X') (so that there is a Y-equivariant homeomorphism from 9X to dX'). Then, there is a compactified space, (Y, dY), admitting a proper Γ-action, and Γequivariant morphisms f : (Y, dY) —» (Χ, 9Χ) and f: (Y, dY) —> (Xr, dX').

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Proof. Choose compact sets K c X and Κ' c Χ' such that X = U Γ AT and X' = U V K'. Leth: dX —> BX' be a Γ-equivarianthomeomoφhism. Let Υ = \J Γ(Κ χ K') and let dY = graph(A). By Lemma 6.3, we see that Υ U 97 is a closed subset of (X U BX) χ (X' U BX'). We see that (7, 37) is a compactified space. Moreover, the coordinate projections, /: Υ U dY —^ X U 3X and /': Υ U dY —> X'UdX' are morphisms. D (In fact, it's not hard to see directly that the conclusion of Corollary 6.4 defines an equivalence relation on the proper actions of a fixed group. We have thus shown that, for any given group, there is just one equivalence class.) Proposition 6.5. Suppose that Γ acts properly on compactified spaces (X, dX) and (X', BX'). Suppose that f : X U 9X —> X' U BX' is α Γ-equivariantfunction with f ( X ) C X' and f\X continuous, and with f \ d X a homeomorphism onto dX'. Then, f is continuous. Proof. Let Κ c X be compact, with X = \JTK. Let Υ = \JT(K χ f K ) U graph(/|3X). Thus, graph(/) c y,andy is a closed subset of (X d X ) x ( X ' dX'). Suppose xn -> χ e dX and f(xn) -»· y e X' U BX'. We see that (x, y) e Υ Π (ΒΧ χ (Xr U 3Χ')) = y n (9Χ χ 3Χ') = graph(/|3X) and so y = f ( x ) . This shows that / is continuous on BX, and hence on X U 3X. D Thus, if /1X is a homeomorphism of X onto X', we get an isomorphism of (Χ, 9Χ) to (X', 9X0· In particular, this proves the uniqueness of compactifications of properly discontinuous cocompact actions. As remarked earlier, it also gives another proof of the uniqueness of the topological conjugacy between two uniform convergence actions — consider the induced actions on the compactified spaces of triples. We need to consider the question of existence of compactifications. To this end, we begin by observing that one can reconstruct compactified spaces as domains or ranges of morphisms. We first make a few general topological observations. Suppose that M is a compactum. The topology on M is unique among comparable topologies in the sense that any strictly coarser topology will fail to be hausdorff and any strictly finer topology will fail to be compact. If N is another compactum, and /: TV —> M is a continuous surjective map, then the topology on M is determined as the quotient topology. In other words it is the coarsest topology such that / is continuous. Alternatively, it is the unique hausdorff topology such that / is continuous. Note that if U c M then the subspace topology on U is the quotient of the subspace topology on / '[/. More generally, if X and Y are locally compact hausdorff spaces, and /: y —> X is a continuous proper surjective map, then the topology on X is also determined as the quotient topology. To see this, note that / extends to a continuous surjective map from the one-point compactification of Y to the one-point compactification of X. The statement follows from the observations of the previous paragraph.

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Suppose, now, we are given a compactified space, (Y, dY), and a continuous proper surjective map, /: Υ —> X, to a locally compact hausdorff space, X (so that the topology on X is the quotient topology from Y). We may extend / to a morphism as follows. We let dX = dY (as a set), and extend / to a map /: Y U dY —> X(JdX by setting f\dY to be the identity. We give X U dX the quotient topology. Clearly X U 3X is compact, and since every point preimage in Y U dY is compact (given that our original map / was proper) we see that X U 3X is also hausdorff. We also see that f \ d X is a homeomorphism, and X is open in X U 3X. Also the new (subspace) topology on X is the quotient topology from K, and thus agrees with the original. Clearly, if Y is dense in Y U dY, then X is dense in X U dX. We conclude that X U dX is a compactified space, and /: (Y, dY) —> (X, 3X) is a morphism. Moreover, (Χ, 3Χ) is unique up to isomorphism. Now, suppose we are given a compactified space, (X, 3X), and a continuous proper surjective map, /: Y —> X. As before, we want to extend / to a morphism. Again, we set dY = dX (as a set) and extend / to Y U dY by taking f\dY to be the identity. We topologise Y U dY by taking as base all open subsets of Y together with all sets of the form / ' V as V varies over open subsets of X U dX. This collection is clearly closed under finite intersection, and is thus indeed a base for a topology. Moreover, / is continuous, and so X U 3X is hausdorff. Also f\dY is a homeomorphism, and the subspace topology on Y agrees with the original. We need to check that Y U dY is compact. To this end, suppose that V. and V are collections of open subsets of Y and X U 3X respectively, such that U U {/-' V \ V e V} covers Y U dY. Now, V covers 3X, and so there is some finite subset V0 c V covers 3X. Let A: c X\U^o· Thus, A: is a compact subset of X. Since / is proper, /~' K is a compact subset of Υ, and hence of Y U dY. It is thus covered by a subset, UQ (J{f~' V \ V e V\}, where U0 c U and V\ c V are finite. It follows that Y U dY is covered by 1i0 U {/~' V \ V e V0 U V\}. Thus Y U dY is compact. Note that if X is dense in X U 3X, then Y is dense in Y U dY. We have shown that (Y, dY) is a compactified space, and that/: (Y, dY) —> (Χ, 3Χ) is a morphism. In fact, the construction of the last paragraph is natural up to isomorphism. Indeed, the topology on y U 3 Y is determined as the unique compact topology inducing the original topology on Y and such that / is continuous. To see this, suppose that Y U dY admits another topology with this property. It's clear that this topology must be finer that constructed above. However, if it were strictly finer, then it would fail to be compact. Suppose now that Γ acts properly on the compactified space (Χ, 3X), and acts properly discontinuously and cocompactly on a locally compact hausdorff space, X'. We can compactify X' as a space (Χ', dX'), admitting a proper Γ-action, as follows. Let Κ C X and K' c X' be compact sets such that X = jj Γ Κ and X' = \J ΓΚ'. Let 7 = U Γ (Κ χ Κ') c Χ χ Χ', and let /: Υ —» Χ and /': Υ —* Χ' be the natural projection maps. Now, Y is closed in Χ χ Χ', and hence locally compact. Moreover, / and /' are proper and surjective. As described earlier, we can find a compactified space (Y, dY) and extend / to a morphism /: (Y, dY) —> (Χ, 3Χ).

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Since this construction is natural, we get a Γ-action on (Y, dY) such that the map / is Γ-equivariant. By Lemma 6.2, this action is proper. Similarly, we construct a compactified space (X', dX') admitting a proper Γ-action, and extend /' to a Γequivariant morphism /': (Y, dY) —> (Xf, dX'). Now given any uniform convergence action of a group, Γ, on a perfect compactum, M, we can take, as starting point for the above construction, the proper action of Γ on the compactified space (Θ(Μ), θ Θ (Λ/)). We conclude: Proposition 6.6. Suppose we are given a uniform convergence action of a group Γ on a perfect compactum, M. Suppose Γ acts properly discontinuously cocompactly on a locally compact hausdorff space X. Then there is a natural compactification of X as a compactified space (X,dX) admitting a proper action of the group Γ, extending the action of Γ on X. This compactification is unique up to isomorphism. Moreover, there is a unique Γ -equivariant homeomorphism of M onto dX. Note that if M is disconnected, then Γ has more than one end. This follows, either from Proposition 6.6, (taking X to be the Cayley graph), or appealing to [Bo6] and the standard fact for hyperbolic groups. Thus, Stallings's theorem [St] tells us that Γ splits over a finite subgroup. Dunwoody's accessibility theorem [Dl] then leads us naturally to considering uniform convergence actions on continua.

7. Configuration spaces in continua In this section, we give some general results relating to continua (connected compacta). Our main concern will be with connectedness properties of configuration spaces. For applications, this means spaces of triples, though for the most part, we have little reason to restrict to this case. From our discussion we can deduce something about convergence groups acting on continua. We shall proceed here in a fairly general manner, given what seem to be some useful general observations along the way. (For some general discussion of the theory of continua, see, for example, [HY] and [Na].) Suppose that M is any hausdorff topological space. Let Π^ (Μ) c Mn be the open subset of distinct ordered η-tuples (i.e. η-tuples with no two entries equal). Let Un(M~) be the quotient space of Π®(Μ) under the action of the symmetric group on η letters which permutes the coordinates. We think of an element of Un(M) as an unordered η-tuple, in other words a subset of M of cardinality n. We refer to the spaces Π®(Μ) and Πη(Μ) as "configuration spaces". Note that Θ°(Μ) = ΓΤ^Μ) and Θ(Μ) = Π3(Μ). Our main interest is in Θ(Μ), though since there is nothing very special about the number 3, we may as well proceed in greater generality. Our first main result (Theorem 7.3) tells us that if M is connected, then so is n„ (M) for all n. Alhough it is of no direct relevance to the rest of the paper, it is also interesting to consider when Π®(Μ) is connected (in the case of metrisable continua). Finally, if M is a Peano continuum, we shall see that Πη (Μ) has only one end for n>2.

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All of this is clearly related to the manner in which M is separated by finite subsets. This seems to be an interesting question in itself. As a special case we have the "treelike" nature of the set of global cut points of a connected hausdorff space, as formulated in [W]. (See also [Bo2].) More generally, we have the following lemma, which seems to be quite useful, though I haven't found any mention of it in the literature. First, recall that a quasicomponent of a topological space is an equivalence class under the equivalence relation defined by deeming to points to be equivalent if every clopen (closed and open) subset containing one also contains the other. Quasicomponents are always closed (though not necessarily connected). Clearly, a space is connected if and only if it has precisely one quasicomponent. For further discussion, see [HY]. If M is a connected hausdorff space, C c M is closed, and x, y e M \ C, we say that C separates χ from y in Μ if Λ and y lie in different quasicomponents of Μ \ C. In other words, we can write Μ \ C as a disjoint union of two open sets Μ \ C = O u U with χ e Ο and y e U. It turns out that if F c Μ is any finite subset, then the separation properties of (subsets of) F are determined by an embedding of F in a finite graph. To be more precise, we construct a finite graph, G = G(M, F), with vertex set V(G) = F, by joining jc, y e F by an edge if χ and y lie in the same quasicomponent of (M \ F) U {*, y}. Lemma 7.1. Suppose M is a connected hausdorff space, and F C. M is finite. Let G = G (M, F) be the finite graph described above. IfC C F and a, b e F \ C, then C separates a from b in M if and only ifC separates a from b in G. Proof. Suppose a and b are connected by an edge in G. Then a and b lie in the same quasicomponent of (M \ F) U {a, b}, and hence in the same quasicomponent of M \ C. More generally, it follows that if a and b are connected by a path in G \ C, then they lie in the same quasicomponent of M \ C. To prove the converse, suppose that we can write F = A u B u C, with a e A and b e B, and such that no edge of G connects any point of A to any point of B. Suppose χ € A and y € B. By the definition of a quasicomponent, we can write (M \ F) U {*, y} = O U U, with Λ; e Ο, y e U and O, U open. We write O(x, y) for some such set O. Note that its closure, 0(x, y), is contained in (O(x, y) U F) \ {y}. Now, let O(x) = Π ν£ β O(x, y). Thus, O(x) is open, χ € O(x) and Β Π O(x) = 0. Moreover, 0(x) C p|v6 (x, y) C (O(x) U F) \ B = O(x} U (F \ B). Now let Ο = \JxeA 0(x). Thus, Ο is open, A c O, and Β Π Ο = 0. Moreover, 0 C O U (F \ ) = O U (A U C) = O U C. Let U = M \ ( U C). Then, M \ C = O U U, with O, U open, α ε Ο and b € U. Thus, a and b lie in different quasicomponents of M \ C. D Corollary 7.2. G(M, F) is connected. Proof. Take C = 0.

D

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Before applying this to configuration spaces, we show how it gives us the "treelike" nature of cut points in a connected hausdorff space, as alluded to earlier. This treelike structure is critical in obtaining splitting of one-ended hyperbolic group with nonlocally-connected boundary, leading eventually to the conclusion that no such groups can exist. Suppose that M is a connected hausdorff space. Given x,y,z e M, we say that y lies between χ and ζ if y separates χ and ζ in M. This defines a ternary "betweenness" relation on M. It's known that this relation satisfies certain axioms introduced by Ward [W]. These axioms turn out to be equivalent to the following property. Suppose F c M is any finite subset, then we can embed F in a finite tree, T, such that if x,y,z € F, then y lies between χ and ζ in M if and only if y lies between χ and ζ in Γ. In other words, if we restrict Lemma 7.1 to the case where card C < 1, we can suppose that our finite graph is a tree (except that F need not the the entire vertex set of the tree). Deducing this fact from Lemma 7.1 is elementary graph theory. Suppose that G is any finite connected graph. A block in G is a maximal 2-vertex connected subgraph. (We consider a single edge to be 2-vertex connected.) Thus, two blocks intersect, if at all, in a common vertex. Let T(G) be the bipartite graph whose vertex set is an abstract disjoint union of the vertex set of G and the set of blocks of G. An edge of T(G) connects a vertex to a block if and only if the vertex lies in the block (in G). One verifies that T(G) is a tree. Moreover, if x, y, z are vertices of G and hence also of r(G), then y lies between χ and ζ in T(G) if and only of y separates χ from z in G. Thus, starting with F as a finite subset of our space M, and setting G = G(M, F), we see that the betweenness relations on F as a subset of M agree with those on F as a subset of T(G). An alternative proof of the existence of such a tree is given in [Bo2]. The treelike structures arising from these axioms are analysed in that paper, and, from a somewhat different perspective, in [AdN]. We now return to the objective of studying configuration spaces. As before, M is a connected hausdorff space. The following observation will be useful. Fix η > 2, and suppose that C C M is a subset with η — 1 elements. The map [x M» C U {χ}]: M \C —> n„(M) is continuous. In particular, we see that if α and b lie in the same quasicomponent of M \ C, then C U {a} and C U {b} lie in the same quasicomponent of Π η (Μ). We can now prove: Theorem 7.3. If M is a connected hausdorff space, and η > 1, then Π η (Μ) is connected. Proof. Suppose, first, that F C M is a subset with η + 1 elements. Thus, each x € F gives us an element, F \ {x}, of Πη (Μ). Now, if x, y e F are connected by an edge of G = G(M, F), it follows, from the definition of G and the observation immediately preceding the proof, that F \ {x} and F \ {y} lie in the same quasicomponent of Πη (Μ). Since G is connected (by Corollary 6.2), it follows that the elements, F \ {x} lie in the same quasicomponent of Π η (Μ) for all x e F.

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Now, we can get from any set of n elements of M to any other by moving one element at a time. From the previous paragraph, we see that each such move keeps us in the same quasicomponent of Πη(Μ). It follows that n„(M) has only one quasicomponent, and is thus connected. D Although we shall have no need of the result here, it is amusing to ask when the space of distinct ordered «-tuples is connected. If we restrict to the case where M is a metrisable continuum, we can give a complete answer to this question: Proposition 7.4. Suppose M is metrisable continuum not homeomorphic to an interval or a circle, then Π®(Μ) is connected. Clearly, if Μ is a (non-degenerate) interval, then Π η (Μ) has precisely n\ components, whereas if Μ is a circle, it has (n — 1)! components. (In these cases, everything is locally connected, so components and quasicomponents agree.) We shall only give a rough sketch of the argument here. To this end we shall need topological characterisations of circles and finite trees among metrisable continua. (By a "finite tree", we really mean the realisation of a finite simplicial tree). Recall that a (global) cut point in a continuum is a point whose complement is disconnected. The following result can be found in [Na]: Lemma 7.5. A metrisable continuum is a finite tree if and only if it has finitely many non-cut points. A characterisation of the circle is given in [HY]. Thus, Μ is homeomorphic to a circle of and only if the complement of any pair of distinct point of Μ is disconnected. We shall need a slight variation on this, as follows. Given a continuum, M, define a 4-ary relation, 8, on Μ as follows 8(x, y, z, w) holds if and only if the pair {x, z} separates y from w. We say that Μ is cyclically separated if δ is a cyclic order. The following can be deduced from the result cited in the previous paragraph. We omit the proof. Lemma 7.6. A cyclically separated metrisable continuum is homeomorphic to a circle. Now, Lemma 7.1 effectively reduces Proposition 7.4 to a problem in graph theory. Suppose G is a finite connected graph, and I is a set with n elements. Consider the collection of injective maps into the vertex set, V(G), of G. We say that two such maps, /, g: I —>· V(G) are related by a move if there is some / e / such that f(i) and g(i) are adjacent, and f\I \ {/} = g\I \ {/}. Intuitively, we imagine placing counters labelled by the elements of / on distinct vertices of G. We can think of a move as sliding a counter labelled i form one vertex to an adjacent vacant vertex. Suppose A c V(G) is a subset of n elements, and we have two functions /, g which are related by a finite sequence of moves, and with / ( / ) = g ( / ) = A. Then, g o / " 1 gives us a permutation of A. The set of permutations arising in this way defines a subgroup of the symmetric group on n letters which is well defined up to conjugacy,

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and independent of the choice of A. If this subgroup is the whole symmetric group, we say that G is n-permutable. One can ask which graphs have this property. The following result is no doubt far from optimal. We omit the proof. Lemma 7.7. Given any n e J*i, there is some k(n) e 2, and then oce and e are the words xyxy ... and yxyx ..., each of length me. A monoid defined by such a presentation is called an Artin monoid, and these are therefore embeddable in groups provided that either Γ is triangle-free or else each me is at least three. Another nice result for these monoids is that of Deligne [16], who showed that an Artin monoid is embeddable if the corresponding Coxeter group defined by the (group) presentation (A : {a2 : a E A} U {(xy)m' : e = {*, y} E E}) is finite. (The Coxeter group is obtained by making each generator in the presentation for the Artin group have order two.) It is still (we believe) an open question as to whether all Artin monoids are embeddable in groups. One can also consider semigroup analogues of the Coxeter groups, but, unlike the Artin groups which give rise immediately to semigroup presentations, we have to transform the group presentation into a semigroup presentation in this case. One way of doing this is that described in [8]. Here we do not insist that all the group generators have order two, so that we are assigning an order pa to each generator a of the group (or, equivalently, an integer pa > 0 to each vertex a of the graph Γ). In addition, it is more convenient now to think of Γ as a directed graph, so that we are assigning an integer me > 2 to each directed edge e, and we insist that Γ has no directed circuits. Given this, we may form the (semigroup) presentation pr = {A : [apa+l = a : a E A} U {#fl)fc : a, b E A } ) , where (ab)me = ap" (ba}m* = bph ab = ba

ife = (a, b) is an edge ife = (b, a) is an edge otherwise.

We say that a graph Γ is finitely related if G(pr) is a finite group. If Γ is such a graph, we say that a vertex α of Γ is an initial vertex if there is no edge of the form (b,a) in Γ, and we then define strongly finitely related graphs as follows: • every graph with one vertex is strongly finitely related;

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• a graph Γ with η + 1 vertices is strongly finitely related if it is finitely related and if every graph Γ — {a}, where a is an initial vertex in Γ, is strongly finitely related. Given this, the main result of [8] can be summed up as follows: Theorem 4.2. Let Γ be a graph with no directed circuits, with an integer pa > 0 assigned to each vertex a and an integer me > 2 assigned to each directed edge e. Then S(pr) has a unique minimal left ideal which is a union of minimal right ideals, each of which is isomorphic to the group G(pr) via the natural homomorphism from to G(pr)· Moreover, S(pr) is finite if and only if Γ is strongly finitely related. A right ideal in a semigroup S is a subset R such that RS c R, a left ideal is a subset L such that S L c L, and an ideal is a subset which is both a right ideal and a left ideal. In Theorem 4.2, we see that the natural mapping -ψ from 5"(p) to G(p) is surjective, but need not be injective; we shall explore this possibility further in the next section.

5. Surjectivity We have considered the case where the natural mapping ψ from S(p) to G(p) is injective; alternatively, one can ask if ψ maps 5(p) surjectively onto G(p) (as happened in the situation described in Theorem 4.2). Of course, this will not happen in general (for example, in the case of a free semigroup), but it will always happen in a periodic semigroup, i.e. a semigroup S in which each element a has "finite order". (For semigroups, we say that an element a has finite order if there exist 1 < n < m with a" = am.) Surjectivity (in the case where 5(p) is periodic) follows from the fact that, if p is the presentation (A : 91), then the image of ψ is the subsemigroup Τ of G(p) generated by {αψ : a e A}. Since a" = am implies that each a has finite order (in the group sense) in G (p), we see that each a~l is in T, so that T = G. In fact, we can see that we only need each generator of S(p) to have finite order. However, we do not need this assumption for -ψ to be surjective. For example, it is shown in [5] that ψ is necessarily surjective if 5(p) has minimal left and right ideals. In this case, it was already known ([14], [43]) that S = S(p) has a unique minimal ideal K (the kernel of S) which is the union of all the minimal left ideals of 5, and also the union of all the minimal right ideals of 5; in addition, if R is any minimal right ideal of S, and L is any minimal left ideal, then Α Π L is a group. The (isomorphism class of the) group R Π L is independent of the choice of R and L, and we denote this group by //(p) (where //(p) only makes sense if S has minimal left and right ideals). The main result of [5] is then Theorem 5.1. Let p be a presentation, and suppose that S = S ( p ) possesses both minimal left and minimal right ideals (so that S has a kernel K). Let ψ be the natural

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homomorphism from S to G = G (p), let L be a minimal left ideal of S, let R be a minimal right ideal, and let H = //(p) be the group R Π L. Let E denote the set of idempotents in K. Then ψ\π '· H —>· G is a group epimorphism; moreover, •φ· \H '· Η —>· G w a group isomorphism if and only if E is a subsemigroup ofS. By an idempotent, we mean an element e such that e2 = e. We see that H = G if and only if the set E of idempotents in K is closed. A particular case where this must happen is where we have a single minimal left ideal (or a single minimal right ideal), since E is necessarily closed in this case; this happened in the situation described in Theorem 4.2. The kernel A" is a semigroup with minimal left and right ideals but without any proper ideals; such a semigroup is said to be completely simple, and is known to be isomorphic to a Rees matrix semigroup (see [30] for details).

6. Presentations of subsemigroups Given all this, a further natural question to ask is the following: given a finite presentation p = (A : JH), and assuming we do have a minimal left ideal L and a minimal right ideal R in S(p), can we determine a presentation for //(p)? A method for doing this is demonstrated in [6]. Suppose that S(p) has finitely many minimal left ideals and finitely many minimal right ideals (though at least one in each case). Then there is a constructive procedure which, given a word representing an element in the kernel, determines a presentation for H (p) . We will not describe the method here, but sum up the results in the following: Theorem 6.1. Let S be a finitely presented semigroup with finitely many minimal left ideals and finitely many minimal right ideals. Suppose that we are given a word representing an element of some minimal left ideal L. Then the group H = R Π L and the kernel K of S are both finitely presented and there is an algorithm to determine presentations for both H and K. The fact that we can derive a presentation for K once we have found a presentation for H follows from [31]. The method referred to in Theorem 6.1 was used in [6] to give a presentation for //(p) in the Fibonacci semigroups, which are the semigroups defined by the presentations for the Fibonacci groups; for more information about these groups, see [50]. This verified a conjecture stated in [4]. A particular case (by way of illustration) is the following: Theorem 6.2. If p is the presentation

X4X5 = * ι , ΧζΧβ = X2,X6X\ =

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then G(p) is the cyclic group of order 1 while 5(p) is a completely simple semigroup which is a union of six copies of the free group of rank 2. However, Theorem 6. 1 tells us very little in the case where our semigroup has a zero (i.e. an element z such that zx = χ z = z for all χ). If we have such an element, then {z} is the unique minimal left ideal and the unique minimal right ideal, and both the groups //(p) and G(p) are trivial. In this case it makes sense to talk of a 0minimal left ideal, which is a left ideal L φ {ζ} such that there is no left ideal L' with {z} C L' c L. (Note that any left ideal must contain z.) We may analogously define a 0-minimal right ideal and a 0-minimal ideal. Suppose, then, that we have a 0-minimal ideal / with 0-minimal left and right ideals; such an ideal / is a completely 0-simple semigroup. The intersection of a 0minimal left ideal L of I and a 0-minimal right ideal R of / is clearly not a group (as it contains z). However, it is known that (L Π /?) — {z} either has zero multiplication (i.e. xy = z for all χ and v) or else is a group. For every 0-minimal right ideal R, there is at least one 0-minimal left ideal L such that n L is a group, and, again, the (isomorphism class of the) group is independent of the choice of R and L. This group H is often called the Sch tzenberge r group of / (see [33] for example). It was shown in [7] how the methods of [6] can be extended to this case and so we can find a presentation for H here too. Another interesting feature is that it turns out that we can drop the hypothesis that there are both finitely many (O-)minimal left ideals and finitely many (O-)minimal right ideals, in that finitely many of either is sufficient to make H finitely presented. (On the other hand, we still need that our (O-)minimal ideal / contains only finitely many of each for / to be finitely presented.) We sum up our conclusions in the following: Theorem 6.3. Let S be a finitely presented semigroup with a Q-minimal ideal I which is a completely 0-simple semigroup with finitely many 0-minimal left ideals or finitely many Q-minimal right ideals. Then the Schutzenberger group H of I is finitely presented. If there are both finitely many Q-minimal left ideals and finitely many 0minimal right ideals, then I is finitely presented. One might ask to what extent one can generalize this result to Schutzenberger groups of a regular .©-class. (See [33] for the general definition of a Schutzenberger group.)

7. Subsemigroups of finite index The results in Section 6 are interesting in that they give sufficient conditions for certain subsemigroups of S(p) to be finitely presented. However, they do not really correspond in a natural way to group theoretic results, in that we do not have any proper ideals in a group. What we would like here is an analogue of the Reidemeister-Schreier

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results that say that, if we have a finite presentation p, then a subgroup of G(p) of finite index is itself finitely presented (and give a constructive method of finding such a presentation). At first sight, this seems hopeless, since the index of a subgroup in a group is defined in terms of its cosets, and we do not have such a notion for a subsemigroup of a semigroup. In addition, semigroups behave very differently from groups. For example, we have the well known Nielsen-Schreier theorem that a subgroup of a free group is free. In contrast, a subsemigroup of a free semigroup need not be free. In fact, a finitely generated subsemigroup of a free semigroup need not even be finitely presented. We consider an example from [9]. If S is the free semigroup on a, b and c, and if T is the subsemigroup generated by v = ba, w = ba2, χ = α 3 , y = ac and ζ = a2c, then Γ is not finitely presented; in fact, Τ has presentation (v, w, x, y, ζ : vxlz = wxly (i > 0)), and no proper subset of these relations is sufficient to define T. On the other hand, if we consider the free group G on a, b and c, and the subgroup U generated by v, w, x, y and z, then not only is U free, but it is actually equal to G. (However, Spehner has shown that every finitely generated subsemigroup of a free semigroup admits a finite Malcev presentation; see [48] for details.) As far as (right) ideals in a free semigroup are concerned, we have the following result from [9]: Theorem 7.1. Let S be a free semigroup. If I is a proper ideal of S, then I is not free. If R is a proper right ideal of S which is finitely generated as a semigroup, then R is not free. In the case of a group G, if we have a subgroup U of finite index in G, then the largest normal subgroup N of G contained in U also has finite index in G, and the group G/N acts as a finite group of permutations on the cosets of U. The analogous construction for a quotient of a semigroup is that of the Rees quotient, where we take an ideal T and form the semigroup S/ T with elements (S — T} U {z} (where z ^ S — T) and with multiplication Φ defined by • aVb = cifa,b, c e S -T with ab = c in S, • aVb = zifa,beS-T

with ab e T, and

• α φ ζ = zVa = zforallfl. We see that S/ T will be finite if and only if 5—T is finite. We extend this to semigroups by following [32] in saying that a subsemigroup Γ of a semigroup S has finite index in S if S — T is finite. The following result was proved in [32] (see also [7]): Theorem 7.2. IfS is α finitely generated semigroup and T is a subsemigroup of finite index in S, then T is finitely generated.

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Moreover, it was shown in [7] that, if S is finitely presented and T is an ideal of finite index in S, then T is finitely presented. This was generalized in [9] to the case where Γ is a right ideal of finite index in S, and then in [46] to arbitrary subsemigroups, so that we have: Theorem 7.3. IfS is a finitely presented semigroup and Τ is a subsemigroup of finite index in S, then Τ is finitely presented. Some related finiteness questions are discussed in [47]. One application of Theorem 7.3 is to ideals in free semigroups. It was shown in [9] that finitely generated ideals in free semigroups have finite index; since the existence of a finitely generated ideal in a free semigroup implies that the semigroup is finitely generated (and hence finitely presented), Theorem 7.3 implies that such ideals are finitely presented. In fact, it is shown in [9] that we can extend this to finitely generated right ideals (though they need not be of finite index): Theorem 7.4. If S is a free semigroup and R is a right ideal of S which is finitely generated as a subsemigroup, then R is finitely presented. These are situations where finitely generated implies finitely presented; another such case is that of commutative semigroups (see [42] for example). However, as we pointed out above, this does not happen for arbitrary subsemigroups of free semigroups. In general, finitely generated ideals in semigroups need not be finitely presented; see [11] for example.

8. Free products Having discussed free semigroups (among other things) in the last section, we now turn our attention to free products of semigroups. The free product of semigroups S] and 82 with presentations (Ai : 9ij) and (Ai : ^2) (with A\ and A2 disjoint) is the semigroup with presentation (A\ U A 2 : 9ij U 9Ϊ2>, so that the free product is finitely presented if and only the factors are (unlike direct products of semigroups; see [45]). Note that, if S\ and £2 are groups, then this free product is not the same as the group free product; the group free product is, essentially, the semigroup free product amalgamating the trivial subgroup. One consequence is that, while the group free product of a finite group and the trivial group is finite, the semigroup free product of any two semigroups is always infinite. If 5Ί and £2 are trivial semigroups, then it is shown in [ 10] that every subsemigroup is finitely generated and finitely presented; however, the situation changes if either factor is non-trivial: Theorem 8.1. If S\ and 82 are two semigroups with at least one of S\ and 82 nontrivial, then the free product S = S\ * £2 contains a two-sided ideal which is not finitely generated and a finitely generated subsemigroup that is not finitely presented.

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This is in contrast to the case of the (group) free product of groups where we have the Kurosh subgroup theorem. In particular, if G j and G 2 are such that every finitely generated subgroup is finitely presented, then every subgroup of G i * G^ also has this property. We saw in Theorem 8.1 that a finitely generated subsemigroup of a free product of semigroups S\ and £2 need not be finitely presented even if Si and £2 are themselves finitely presented. The situation is quite different if we consider ideals, as the next result from [10] shows: Theorem 8.2. If Si and 82 are two finitely presented semigroups and I is an ideal of S\ * 5*2 which is finitely generated as a subsemigroup, then I is finitely presented. This result is proved by showing that such an ideal 7 must have finite index, and Theorem 8.2 then follows immediately from Theorem 7.3. We do not know whether this can be generalized to right ideals, and we pose: Question 8.3. // S\ and 82 are two finitely presented semigroups and R is a right ideal ofS\ * 82 which is finitely generated as a subsemigroup, is R necessarily finitely presented? The answer to this question is "yes" in the special case where we have a free product of semigroups where each factor is either a free commutative semigroup or a finite semigroup: Theorem 8.4. If S = S\ * 5*2 * · · · * Sn, where each Si is either a free commutative semigroup or a finite semigroup, then every right ideal of S which is finitely generated as a subsemigroup is finitely presented. This was proved in [10]. As every free semigroup of finite rank is a free product of finitely many free monogenic semigroups, Theorem 8.4 generalizes Theorem 7.4. Acknowledgements. Our presentation of this material has benefited from conversations with Jim Howie and Steve Pride over a number of years, and we are very grateful to them for their help; the fourth author would like to thank Hilary Craig for all her help and encouragement. The authors also wish to acknowledge support from the Edinburgh Mathematical Society Centenary Fund and the European Community Grant ERBCHRXCT-930418 which helped to make possible visits of the fourth author to the University of St Andrews while some of the work described in this paper was being undertaken and also to the Edinburgh Mathematical Society Centenary Fund for further support for a visit of the first author to Leicester.

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References [1]

S.I. Adjan, On the embeddability of semigroups in groups, Soviet Math. Dokl. 1, (1961), 819-821; translated from Dokl. Akad. Nauk. SSSR 133 (1960), 255-257.

[2]

S. I. Adjan, Defining relations and algorithmic problems for groups and semigroups, Proc. Steklov Inst. Math. 85 (1966), American Mathematical Society (1967); translated from Trudy. Mat. Inst. Steklov. 85 (1966).

[3]

G. Baumslag, J. W. Morgan and P. B. Shalen, Generalized triangle groups, Math. Proc. Cambridge Philos. Soc. 102 (1987), 25-31.

[4]

C. M. Campbell, E. F. Robertson, N. Ruskuc and R. M. Thomas, Fibonacci semigroups, J. Pure Appl. Algebra 94 (1994), 49-57.

[5]

—, Semigroup and group presentations, Bull. London Math. Soc. 27 (1995), 46-50.

[6]

—, Rewriting a semigroup presentation, Internat. J. Algebra Comput. 5 (1995), 81-103.

[7]

—, Reidemeister-Schreier type rewriting for semigroups, Semigroup Forum 51 (1995), 47-62.

[8]

—, On semigroups defined by Coxeter type presentations, Proc. Royal Soc. Edinburgh 125A(1995), 1063-1075.

[9]

—, On subsemigroups of finitely presented semigroups, J. Algebra 180 (1996), 1-21.

[10] —, On subsemigroups and ideals in free products of semigroups, Internat. J. Algebra Comput. 6(1996), 571-591. [11] —, Presentations for subsemigroups — applications to ideals of semigroups, J. Pure Appl. Algebra 124 (1998), 47-64. [12] C. M. Campbell, E. F. Robertson, N. Ruskuc, R. M. Thomas and Y. Unlii, On certain one-relator products of semigroups, Comm. Algebra 23 (1995), 5207-5219. [13] J. R. Cho and S. J. Pride, Embedding semigroups into groups and the asphericity of semigroups, Internat. J. Algebra Comput. 3 (1993), 1-13. [14] A. Clifford, Semigroups containing minimal ideals, Amer. J. Math. 70 (1948), 521-526. [15] D. J. Collins and J. Perraud, Cohomology and finite subgroups of small cancellation quotients of free products, Math. Proc. Cambridge Philos. Soc. 97 (1985), 243-259. [16] P. Deligne, Les immeubles des groups de tresses generalises, Invent. Math. 17 (1972), 273-302. [17] A. J. Duncan and J. Howie, One relator products of high-powered relations, in: Geometric Group Theory Volume 1 (G. A. Niblo and M. A. Roller, eds.), London Math. Soc. Lecture Note Ser. 181, 48-74, Cambridge University Press, 1993. [18] M. Edjvet, An example of an infinite group, in: Discrete Groups and Geometry (W. J. Harvey and C. Maclachlan, eds.), London Math. Soc. Lecture Note Series 173, 66-74, Cambridge University Press, 1992. [19] —, On certain quotients of the triangle groups, J. Algebra 169 (1994), 367-391. [20] M. Edjvet and J. Howie, On the abstract groups (3, n, p; 2), J. London Math. Soc. 53 (1996), 271-288.

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[21] M. Edjvet and R. M. Thomas, The groups (l,m\n, k), J. Pure Appl. Algebra 114 (1997), 175-208. [22] F. Gonzales-Acuna and H. Short, Knot surgery and primeness, Math. Proc. Cambridge Philos. Soc. 99 (1986), 89-102. [23] B. Fine, J. Howie and G. Rosenberger, One-relator quotients and free products of cyclics, Proc. Amer. Math. Soc. 102 (1988), 249-254. [24] P. M. Higgins, Techniques of Semigroup Theory, Oxford University Press, 1992. [25] J. Howie, The quotient of a free product of groups by a single high-powered relator I. Pictures. Fifth and higher powers, Proc. London Math. Soc. 59 (1989), 507-540. [26] —, The quotient of a free product of groups by a single high-powered relator II. Fourth powers, Proc. London Math. Soc. 61 (1990), 33-62. [27] —, The quotient of a free product of groups by a single high-powered relator III. The word problem, Proc. London Math. Soc. 62 (1991), 590-606. [28] J. Howie, V. Metaftsis and R. M. Thomas, Finite generalized triangle groups, Trans. Amer. Math. Soc. 347 (1995), 3613-3623. [29] J. Howie and R. M. Thomas, The groups (2, 3, p; q); asphericity and a conjecture of Coxeter, J. Algebra 154 (1993), 289-309. [30] J. M. Howie, Fundamentals of Semigroup Theory, Oxford University Press, 1995. [31] J. M. Howie and N. Ruskuc, Constructions and presentations for monoids, Comm. Algebra 22 (1994), 6209-6224. [32] A. Jura, Determining ideals of finite index in a finitely presented semigroup, Demonstratio Math. 11(1978), 813-827. [33] G. Lallement, Semigroups and Combinatorial Applications, John Wiley, 1979. [34] L. Levai, G. Rosenberger and B. Souvignier, All finite generalized triangle groups, Trans. Amer. Math. Soc. 347 (1995), 3625-3627. [35] R. C. Lyndon, On Dehn's algorithm, Math. Ann. 166 (1966), 208-228. [36] R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Ergeb. Math. Grenzgeb. 89, Springer-Verlag, 1977. [37] W. Magnus, Über diskontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz), J. Reine Angew. Math. 163 (1930), 141-165. [38] —, Das Identitätsproblem für Gruppen mit einer definierenden Relation, Math. Ann. 106 (1932), 295-307. [39] A. Yu Ol'shanskii, Geometry of Defining Relations in Groups, Nauka, Moscow, 1989. [40] S. J. Pride, Identities among relations of group presentations, in: Group Theory from a Geometric Viewpoint (E. Ghys, A. Haefliger and A. Verjovsky, eds.), 687-717, World Scientific Publishing, 1991. [41] —, Geometric methods in combinatorial semigroup theory, in: Semigroups, Formal Languages and Groups (J. Fountain, ed.), NATO ASI Series C, Mathematical and Physical Sciences 466, 215-232, Kluwer, 1995.

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[42] L. Redei, The Theory of Finitely Generated Commutative Semigroups, Pergamon Press, 1965. [43] D. Rees, On semi-groups, Proc. Cambridge Philos. Soc. 36 (1940), 387^00. [44] J. H. Remmers, On the geometry of semigroup presentations, Adv. Math. 36 (1980), 283-296. [45] E. F. Robertson, N. Ruskuc and J. Wiegold, Generators and relations of direct products of semigroups, preprint. [46] N. Ruskuc, On large subsemigroups and finiteness conditions of semigroups, submitted. [47] N. Ruskuc and R. M. Thomas, Syntactic and Rees indices of subsemigroups, submitted. [48] J. C. Spehner, Every finitely generated submonoid of a free monoid has a finite Malcev's presentation, J. Pure Appl. Algebra 58 (1989), 279-287. [49] J. R. Stallings, A graph-theoretic lemma and group embeddings, in: Combinatorial Group Theory and Topology (S. Gersten and J. R. Stallings, eds.), 145-155, Princeton University Press, 1987. [50] R. M. Thomas, The Fibonacci groups revisited, in: Proceedings of Groups St Andrews 1989 Volume 2 (C. M. Campbell and E. F. Robertson, eds.), London Math. Soc. Lecture Note Series 160, 445^54, Cambridge University Press, 1991.

Conformal modulus: the graph paper invariant or the conformal shape of an algorithm /. W. Cannon, W. J. Floyd, and W. R. Parry

0. Introduction This paper is an expository paper about our joint work, which the first author presented in a series of lectures at the University of Auckland (New Zealand), the University of Melbourne (Australia), and the Australian National University in Canberra (Australia). We express appreciation for the kindness and interest of all the many wonderful mathematicians and their families whom that author and his wife enjoyed during their visit. This final version of the paper includes a few of the questions and comments which arose during the discussions of those lectures. We thank the referees for numerous insightful comments. The first section, which is our own nonproof of the Riemann Mapping Theorem, can be used as a good intuitive introduction to the long and fussy proof of our own combinatorial Riemann mapping theorem [CRMT]. In particular, it demonstrates the geometry underlying the classical conformal modulus of a quadrilateral or annulus. The second section shows how the classical conformal modulus is applied to combinatorics, with the intent of preparing for the exposition of Sections 3 and 4. The third section shows that, under subdivision, a topological quadrilateral can develop wildly oscillating conformal modulus, a behavior which was perhaps not expected. The fourth section reviews how combinatorial moduli apply to the study of negatively curved or Gromov word hyperbolic groups and shows by example how our work might be used to recognize a Kleinian group combinatorially. The final section, Section 5, concludes the paper with remarks and questions.

1. Conformal moduli What is the geometry underlying the modulus formula, Mp = ( H p ) 2 / A p , This work was supported in part by NSF research grants.

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which comes from the theory of conformal mapping and gives the modulus Mp as a ratio which compares the square (Hp)2 of a certain length Hp to an area ΑΡΊ Just as Dr. Strangelove came to love the bomb, so we have come to love this unintuitive expression. Our aim is to explain to topologists and geometric group theorists its beautiful underlying geometry and applications. To which length and area do the symbols Hp and Ap refer? In the sequel we shall consider the formula in four different settings: the classical modulus of a quadrilateral or annulus with a fixed Riemannian metric p\ the conformal modulus of a quadrilateral or annulus obtained by optimizing the classical moduli over a family of Riemannian metrics; the combinatorial modulus of a tiled quadrilateral or annulus with a fixed weight function p\ and the combinatorial conformal modulus of a tiled quadrilateral or annulus obtained by optimizing over a family of weight functions. The first two settings are considered in this section, and the next two settings are considered in Section 2. The remainder of the paper is then devoted to applications of the combinatorial modulus to geometry and group theory. Setting I. The classical, continuous setting. Let Q denote either a (compact) topological quadrilateral (disk with four distinguished boundary points) or a (compact) topological annulus, with Q having a Riemannian metric p. Call two opposite edges of the boundary of Q the top and bottom of Q. (In the quadrilateral case, the four distinguished points of the boundary of Q divide this boundary into four edges, two forming top and bottom, the other two forming the sides. In the annulus case, the two boundary curves of Q are considered opposite edges, the top and bottom of the annulus.) The top and bottom are also called the ends of Q. Then Hp denotes the Riemannian distance between the top and bottom of Q and Ap denotes the Riemannian area of Q. It is easy to understand the geometric meaning of Mp in the case where Q is, as a topological quadrilateral, a true Euclidean rectangle or Q, as an annulus, has the shape of a right circular cylinder. Then top and bottom have obvious geometric meaning, and the distance Hp between top and bottom is the geometric height of Q. The rectangle or right circular cylinder Q has area Ap which is the product of Hp with the width or circumference Wp of Q. Thus Mp is the ratio (HP~)2/AP = HP/WP which obviously measures the geometric proportions or shape of the rectangle or cylinder. See Figure 1. Thus we see that the modulus Mp is a generalized measure of the shape of the quadrilateral or annulus Q, This measure of shape is obviously invariant under scaling of the metric ρ since the height Hp scales by the given scale factor and the area by the square of that same factor. It is precisely this invariance under scaling that dictates the powers of Hp and A p used in the formula. Setting II. The conformal setting. A conformal change of metric multiplies a given metric on Q not by a global scale factor but by an infinitesimal scale factor. The Riemannian metric of Q, which we have been calling ρ up to this point, we now assume fixed and put it into the background without any explicit name. We now reinterpret the symbol ρ as denoting not a Riemannian metric on Q but rather a positive function on Q which serves as the local scale factor of a conformal change of metric. The product of ρ with our fixed but unnamed Riemannian metric on Q gives a new Riemannian metric on , conformally

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Conformal modulus: the graph paper invariant

τ L

R

7 U "p

Hp

Topological quadrilateral

=

HpHp

'·

l

\/

B S , and moduli Mp and Mp'. It will suffice to prove that Mp < K^MP+ = Λ+ χ JCA/ ~

110

RuthChamey

where ( a [ , x \ ) ~ («2,^2) if *i = X2, Xi lies in the relative interior of KT, and [α\\τ = [α2\τ· As described in the previous section, we can decompose K\ into cubes and give each cube a Euclidean metric to obtain a piecewise Euclidean structure on &\. Lemma 3.5. If χ lies in the relative interior of KT, a e A+, and ζ is the image of (a,x) in 3)^, then link(z, 3)A) is isometric to the orthogonal join oflink(x, KT) and

Proof. If α = 1, then the proof is the same as that of Lemma 2.2. Suppose α φ 1 and write \_a~\T = aoAj = [flolr as in Lemma 3.4. Note that the star of ζ (i.e., the union of cells containing z) lies in the subcomplex of £)~^ spanned by St[a0]T = {[b]R | [a0]r < [b]R or [b]R < [a0]T}. Left multiplication by Q > [b] R ι-> [dob] R > defines an order-preserving map Α+-8? which maps 5ί[1]τ isomorphically onto St[ao]r· Thus, the star of ζ = (a, x) is isometric to the star of (1 , x). D

4. Injectivity for two-dimensional groups The natural map θ : A+ ->· A induces a map Θ : · DA which maps α χ isometrically to θ (ά) χ ΧΑ- Since the vertices [a]0, a e A+ are distinct in £>^> to prove that θ is injective, it suffices to prove that Θ is an embedding. Since the positive monoid of a finite type Artin group injects, the natural map £^ -» BAT is an embedding for each AT e & . In light of Lemmas 2.2 and 3.5, Θ is thus a local embedding. To prove injectivity, we will need to show that Θ is a global embedding. This will follow (in the case of a 2-dimensional Artin group) from the next two propositions. Proposition 4.1. Suppose that for every AT e & , &AT satisfies the following property. Ifx, y are two points of distance d(x, y ) < π in £AT andx, y both lie in £}^ , then the geodesic between them also lies in £^ . Then Θ : )^ -^ £>A takes local geodesies to local geodesies. Proof. Recall that a piecewise geodesic γ is a local geodesic if, at the endpoint Λ:,· of each geodesic piece, the incoming and outgoing tangent vectors to γ represent points of distance at least π in the link of ;c,. Thus to show that Θ takes local geodesies to local geodesies, it suffices to show that the induced map link(;t, 3)^) —> link(0(;t), £>A) takes points of distance > π to points of distance > π. By Lemma 2.2 and 3.5

Injectivity of the positive monoid for some infinite type Artin groups

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and properties of orthogonal joins ([CD2], Appendix), this holds if and only if the embedding £~^T ^ &AT takes points of distance > π to points of distance > π. This is precisely what property (P) guarantees. D Remark. If \T\ — 1, 33 AT is 0-dimesional. In this case, we define the distance between any two points in 33 AT to be π and condition (P) is satisfied vacuously. Proposition 4.2. If (Λ, T) is a 2- generator finite type Artin system, then A (= AT) satisfies property (P). The proof of this proposition will occupy the remainder of this section. For more about 2-generator Artin groups see [AS]. Let F(T) denote the free group on T . For w € F(T), let w denote the image of w in A. Suppose w = s"} . . . snkk with s, € Γ, 5,· 7^ $(+ 1 , and η, Φ 0. Define

\\w\\ =k.

The former is called the length of w and the latter is called the syllable length of w. We say a word u e F(T) is an initial substring of w if w is the concatenation of u with some word v. (That is, we require w = uv where no cancellation is possible between u and υ.) Now let Γ = {s, t] and let

A = (s, t | sts . . . = tst . . .} m terms

m terms

with m < oo. Since Λ is finite type, we may view A+ as a subset of A. The simplicial complex 38 A is 1-dimensional with all 1-simplicies of length π/ηι. The 1-simplices are indexed by the elements of A and two such, a and b, have a common vertex if and only if a = bt" or a = bs" for some n e Z. Si^ is the subcomplex of 1-simplices corresponding to a e A+. A piecewise geodesic in £A (between two vertices) is given by a sequence of edges, a\ , . . . , ak such that a, differs from a / + i by a power of a generator. The length of this piecewise geodesic is ^. To prove property (P), it suffices to consider the case where χ and y are vertices in 3$^· (The minimal length counterexample to (P), if any, will occur when Λ: and y are vertices.) Suppose a\ , . . . , «jt is a geodesic from* to y in £A of length less than π (i.e., k < m). To prove Proposition 4.2, we must show that a, lies in A+ for ι = 1, . . . , k. Let Q be an edge in £^ containing χ and let ajt+i be an edge in £^ containing y. Since a, and α,_ι share a vertex for / = 1, . . . , k + 1, we can write a, = aj-\s"' with s, e {s, t}, and we may assume that 5,· ^ s/+i, n/ 7^ 0. Let tu = s"1 . . . s^1 , so ||u»|| = k + 1 < m. Then «ο £ ^ + > «0^ = #/t+i £ ^ + > and for any /, a, = aow for some initial substring M of u;. Hence Proposition 4.2 will follow immediately from the next lemma.

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Lemma 4.3. Let wbea (reduced) word in F(T) with \\w\\ < m and let u be an initial substring ofw. If p e A+ is such that pw lies in A+, then pu also lies in A+. To prove Lemma 4.3, we will need some technical facts about finite type Artin groups in general and 2-generator Artin groups in particular. For two elements a, b e A+ we write a ±i b if a v/ b = 1, that is, if the only c e A+ with c · (2) =>· (3) => (4) is obvious. Assuming (4), we have a = α\Δα2 = Δα\α2 for some a\, ai e A+ and (1) follows. D Lemma 4.6. Let w — σ^σ22 · · · ση"> with σί £ ^-o, £i — ±1, and set k = #{i \ €i = — 1}. Then ΔΙ(ιν lies in A+. Proof. Using the equation Δσ, = σ, Δ we can "slide" copies of Δ to the right until one copy of A precedes each σ, (or σ/) with e, = — 1. Then the equation Δσ(~~ = σ* shows that Akw is a product of elements of MQ, hence lies in A+. D For a word w e F(T)\ets(w) denote the first letter of w and let e(w) denote the last letter of w, without regard to sign. (For example, if w = t2s l, then s(w) = t and e(w) = s.) Suppose w is a reduced word which does not contain any of the words Δ^/ , Δ,^1 as a subword. Then we can factor w into a product of subwords, w = σ^σ^2 . . . σ^", 6,· = ±1, σ/ e MQ which satisfy (4.7)

e(al') = s(a)

if and only if

€i

=

Now let k = #{/ | €{ · — — 1} and let w\, W2 be the subwords of w defined by w\ = σ ' . . . ak and W2 =