Torsion-free Hyperbolic Group Research Articles

Erratum to: “Elementary embeddings in torsion-free hyperbolic groups”

The notations adopted are those of [Per11]. Proposition 5.11 of [Per11] states that if a torsion-free hyperbolic group A admits a cyclic JSJ-like decomposition Λ, and a non injective morphism f : A → A which restricts to conjugation on each non surface type vertex group, and sends surface type vertex groups to non abelian images, then there is a retraction r : A→ A′ which gives A a structure of hyperbolic floor over A′. Unfortunately, we realised that Proposition 5.11 fails to hold in a few exceptional low complexity cases. The natural modification to overcome this mistake is to proceed to a slight generalization of the notion of hyperbolic floors and hyperbolic towers, which we present in Section 1. As we will see in Section 2, however, this does not affect Theorem 1.2, the main result of the paper. Moreover, Theorem 1.2 is the only result which directly uses Proposition 5.11 in its proof. For a corrected version of the paper, see [Per09]. We sincerely apologize for any confusion caused by this mistake.

Effective embedding of residually hyperbolic groups into direct products of extensions of centralizers

Abstract.For any torsion-free hyperbolic group Γ and any group

Algebraic and definable closure in free groups

In Chapter 1 we give basics on combinatorial group theory, starting from free groups and proceeding with the fundamental constructions: free products, amalgamated free products and HNN extensions. We outline a synthesis of Bass-Serre theory, preceded by a survey on Cayley graphs and graphs of groups. After proving the main theorem of Bass-Serre theory, we present its application to the proof of Kurosh subgroup theorem. Subsequently we recall main definitions and properties of hyperbolic spaces. In Section 1.4 we define algebraic and definable closures and recall a few other notions of model theory related to saturation and homogeneity. The last section of Chapter 1 is devoted to asymptotic cones. In Chapter 2 we prove a theorem similar to Bestvina-Paulin theorem on the limit of a sequence of actions on hyperbolic graphs. Our setting is more general: we consider Bowditch-acylindrical actions on arbitrary hyperbolic graphs. We prove that edge stabilizers are (finite bounded)-by-abelian, that tripod stabilizers are finite bounded and that unstable edge stabilizers are finite bounded. In Chapter 3 we introduce the essential notions on limit groups, shortening argument and JSJ decompositions. In Chapter 4 we present the results on constructibility of a torsion-free hyperbolic group from the algebraic closure of a subgroup. Also we discuss constructibility of a free group from the existential algebraic closure of a subgroup. We obtain a bound to the rank of the algebraic and definable closures of subgroups in torsion-free hyperbolic groups. In Section 4.2 we prove some results about the position of algebraic closures in JSJ decompositions of torsion-free hyperbolic groups and other results for free groups. Finally, in Chapter 5 we answer the question about equality between algebraic and definable closure in a free group. A positive answer has been given for a free group F of rank smaller than 3. Instead, for free groups of rank strictly greater than 3 we found some counterexample. For the free group of rank 3 we found a necessary condition on the form of a possible counterexample.

ON ENDOMORPHISMS OF TORSION-FREE HYPERBOLIC GROUPS

Let H be a torsion-free δ-hyperbolic group with respect to a finite generating set S. From the main result in the paper, Theorem 1.2, we deduce the following two corollaries. First, we show that there exists a computable constant [Formula: see text] such that, for any endomorphism φ of H, if φ(h) is conjugate to h for every element h ∈ H of length up to [Formula: see text], then φ is an inner automorphism. Second, we show a mixed (conjugate/non-conjugate) version of the classical Whitehead problem for tuples is solvable in torsion-free hyperbolic groups.

HOMOGENEITY AND PRIME MODELS IN TORSION-FREE HYPERBOLIC GROUPS

We show that any nonabelian free group F of finite rank is homogeneous; that is for any tuples , having the same complete n-type, there exists an automorphism of F which sends ā to .We further study existential types and show that for any tuples , if ā and have the same existential n-type, then either ā has the same existential type as the power of a primitive element or there exists an existentially closed subgroup E(ā) (respectively ) of F containing ā (respectively ) and an isomorphism with .We will deal with non-free two-generated torsion-free hyperbolic groups and we show that they are ∃-homogeneous and prime. In particular, this gives concrete examples of finitely generated groups which are prime and not quasi axiomatizable, giving an answer to a question of A. Nies.

On hyperbolic groups with spheres as boundary

Let G be a torsion-free hyperbolic group and let n � 6 be an integer. We prove that G is the fundamental group of a closed aspherical manifold if the boundary of G is homeomorphic to an (n − 1)-dimensional sphere.

Elementary embeddings in torsion-free hyperbolic groups

The purpose of this thesis is to describe first-order logic elementary embeddings in a torsion-free hyperbolic group. The main result gives such a description in terms of a structure that Sela introduced in his answer to Tarski's problem: the structure of hyperbolic tower. Thus, if H embedds elementarily in a torsion free hyperbolic group G, the group G can be obtained by successive amalgamations of groups of surfaces with boundary to a free product of H with some free group and groups of surfaces without boundary. This gives as a corollary that an elementary subgroup of a finitely generated free group is a free factor. The techniques used to obtain this description are mostly geometric, as for example actions on real or simplicial trees, or the existence of JSJ splittings. We also rely on the existence of factor sets, namely the fact that for some finitely generated groups A, given a torsion-free hyperbolic group G, there exists a finite set of quotients of A such that any non-injective morphism from A to G factors through one of the corresponding quotient maps up to precomposition by an automorphism of A. We give a full proof of these results, including a detailed version of Rips and Sela's shortening argument. The shortening argument shows, using Rips' analysis of actions on real trees, that if a sequence of actions of A on hyperbolic spaces converges to a real A-tree of a specific type, then infinitely many of the actions can be shortened.

Conjugacy classes of solutions to equations and inequations over hyperbolic groups

We study conjugacy classes of solutions to systems of equations and inequations over torsion-free hyperbolic groups, and describe an algorithm to recognize whether or not there are finitely many conjugacy classes of solutions to such a system. The class of immutable subgroups of hyperbolic groups is introduced, which is fundamental to the study of equations in this context. We apply our results to enumerate the immutable subgroups of a torsion-free hyperbolic group.

K-Free-like groups

The following results are proved. In Theorem 1, it is stated that there exist both finitely presented and not finitely presented 2-generated nonfree groups which are k-free-like for any k ? 2. In Theorem 2, it is claimed that every nonvirtually cyclic (resp., noncyclic and torsion-free) hyperbolic m-generated group is k-free-like for every k ? m + 1 (resp., k ? m). Finally, Theorem 3 asserts that there exists a 2-generated periodic group G which is k-free-like for every k ? 3.

Diophantine geometry over groups VII: The elementary theory of a hyperbolic group

This paper generalizes our work on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group, to a general torsion-free (Gromov) hyperbolic group. In particular, we show that every definable set over such a group is in the Boolean algebra generated by AE sets, prove that hyperbolicity is a first-order invariant of a finitely generated group, and obtain a classification of the elementary equivalence classes of torsion-free hyperbolic groups. Finally, we present an effective procedure to decide if two given torsion-free hyperbolic groups are elementarily equivalent.

The isomorphism problem for toral relatively hyperbolic groups

We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorphism problem for finite volume hyperbolic n-manifolds, for n≥3. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsion-free relatively hyperbolic group with abelian parabolics is algorithmically constructible.

Independence Property and Hyperbolic Groups

AbstractIn continuation of [JOH04, OH07], we prove that existentially closed CSA-groups have the independence property. This is done by showing that there exist words having the independence property relative to the class of torsion-free hyperbolic groups.

On superstable CSA-groups

We prove that a nonabelian superstable CSA-group has an infinite definable simple subgroup all of whose proper definable subgroups are abelian. This imply in particular that the existence of nonabelian CSA-group of finite Morley rank is equivalent to the existence of a simple bad group all whose definable proper subgroups are abelian. We give a new proof of a result of Mustafin and Poizat [E. Mustafin, B. Poizat, Sous-groupes superstables de S L 2 ( K ) (submitted for publication)] which states that a superstable model of the universal theory of nonabelian free groups is abelian. We deduce also that a superstable torsion-free hyperbolic group is cyclic. We close the paper by showing that an existentially closed CSA ∗-group is not superstable.

The outer space of a free product

We associate a contractible ‘outer space’ to any free product of groups G = G1 * … * Gq. It is identical to Culler–Vogtmann space when G is free, and McCullough–Miller space when no Gi is ℤ. Our proof of contractibility (given when G is not free) is based on Skora's idea of deforming morphisms between trees. Using the action of Out(G) on this space, we show that Out(G) has finite virtual cohomological dimension, or is VFL (it has a finite index subgroup with a finite classifying space), if the groups Gi and Out(Gi) have similar properties. We deduce that Out(G) is VFL if G is a torsion-free hyperbolic group, or a limit group (finitely generated fully residually free group).

Growth exponent of generic groups

In [GrH97], Grigorchuk and de la Harpe ask for conditions under which some group presentations have growth rate close to that of the free group with the same number of generators. We prove that this property holds for a generic group (in the density model of random groups). Namely, for every positive \epsilon , the property of having growth exponent at least 1-\epsilon (in base 2m-1 where m is the number of generators) is generic in this model. In particular this extends a theorem of Shukhov [Shu99]. More generally, we prove that the growth exponent does not change much through a random quotient of a torsion-free hyperbolic group.

Limits of (certain) CAT(0) groups, I: Compactification

The purpose of this paper is to investigate torsion-free groups which act properly and cocompactly on CAT(0) metric spaces which have isolated flats, as defined by Hruska. Our approach is to seek results analogous to those of Sela, Kharlampovich and Miasnikov for free groups and to those of Sela (and Rips and Sela) for torsion-free hyperbolic groups. This paper is the first in a series. In this paper we extract an R-tree from an asymptotic cone of certain CAT(0) spaces. This is analogous to a construction of Paulin, and allows a great deal of algebraic information to be inferred, most of which is left to future work.

CONJUGACY OF FINITE SUBSETS IN HYPERBOLIC GROUPS

There is a quadratic-time algorithm that determines conjugacy between finite subsets in any torsion-free hyperbolic group. Moreover, in any k-generator, δ-hyperbolic group Γ, if two finite subsets A and B are conjugate, then x-1 Ax = B for some x ∈ Γ with ǁxǁ less than a linear function of max {ǁγǁ : γ ∈ A ∪ B}. (The coefficients of this linear function depend only on k and δ.) These results have implications for group-based cryptography and the geometry of homotopies in negatively curved spaces. In an appendix, we give examples of finitely presented groups in which the conjugacy problem for elements is soluble but the conjugacy problem for finite lists is not.

Automorphisms of Hyperbolic Groups and Graphs of Groups

Using the canonical JSJ splitting, we describe the outer automorphism group Out(G) of a one-ended word hyperbolic group G. In particular, we discuss to what extent Out(G) is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups Out(G) is infinite. We also show that there are only finitely many conjugacy classes of torsion elements in Out(G), for G any torsion-free hyperbolic group. More generally, let Γ be a finite graph of groups decomposition of an arbitrary group G such that edge groups G e are rigid (i.e. Out(G e ) is finite). We describe the group of automorphisms of G preserving Γ, by comparing it to direct products of suitably defined mapping class groups of vertex groups.

Cogrowth and spectral gap of generic groups

The cogrowth exponent of a group controls the random walk spectrum. We prove that for a generic group (in the density model) this exponent is arbitrarily close to that of a free group. Moreover, this exponent is stable under random quotients of torsion-free hyperbolic groups.

Genericity, the Arzhantseva-Ol?shanskii method and the isomorphism problem for one-relator groups

We show that the isomorphism problem is solvable in at most exponential time for a class of one-relator groups which is exponentially generic in the sense of Ol’shanskii. This is obtained by applying the Arzhantseva-Ol’shanskii graph minimization method to prove the general result that for fixed integers m≥2 and n≥1 there is an exponentially generic class of non-free m-generator n-relator groups with the property that there is only one Nielsen equivalence class of m-tuples which generate a non-free subgroup. In particular, every m-generated subgroup in such a generic group G is either free or is equal to G itself and such groups are thus co-Hopfian. These results are obtained by elementary methods without using the deep results of Sela about co-Hopficity and the isomorphism problem for torsion-free hyperbolic groups.