Non-elementary Hyperbolic Group Research Articles

Manhattan geodesics and the boundary of the space of metric structures on hyperbolic groups

For any non-elementary hyperbolic group \Gamma , we find an outer automorphism invariant geodesic bicombing for the space of pseudometric structures on \Gamma equipped with a symmetrized version of the Thurston metric on Teichmüller space. We construct and study a boundary for this space and show that it contains many well-known pseudometrics. As corollaries we obtain results regarding continuous extensions of translation length functions to the space of geodesic currents and settle a conjecture of Bonahon in the negative.

Faithful invariant random subgroups in acylindrically hyperbolic groups

AbstractBuilding on work from Sun and Kechris‐Quorning, we prove that every acylindrically hyperbolic group admits a weakly mixing probability measure preserving action which is faithful but not essentially free. In other words, admits a weakly mixing nontrivial faithful IRS. We also prove that every nonelementary hyperbolic group admits a characteristic random subgroup with the same properties.

Limit Set of Branching Random Walks on Hyperbolic Groups

AbstractLet Γ be a nonelementary hyperbolic group with a word metric d and ∂Γ its hyperbolic boundary equipped with a visual metric for some parameter . Fix a superexponential symmetric probability μ on Γ whose support generates Γ as a semigroup, and denote by ρ the spectral radius of the random walk Y on Γ with step distribution μ. Let ν be a probability on with mean . Let be the branching random walk on Γ with offspring distribution ν and base motion Y , and let be the volume growth rate for the trace of . We prove for that the Hausdorff dimension of the limit set Λ , which is the random subset of consisting of all accumulation points of the trace of , is given by . Furthermore, we prove that is almost surely a deterministic, strictly increasing, and continuous function of , is bounded by the square root of the volume growth rate of Γ , and has critical exponent 1/2 at in the sense that for some positive constant C. We conjecture that the Hausdorff dimension of Λ in the critical case is almost surely. This has been confirmed on free groups or the free product (by amalgamation) of finitely many finite groups equipped with the word metric d defined by the standard generating set. © 2022 Wiley Periodicals LLC.

Random walks on groups and KMS states

A classical construction associates to a transient random walk on a discrete group \(\Gamma \) a compact \(\Gamma \)-space \(\partial _M \Gamma \) known as the Martin boundary. The resulting crossed product \(C^*\)-algebra \(C(\partial _M \Gamma ) \rtimes _r \Gamma \) comes equipped with a one-parameter group of automorphisms given by the Martin kernels that define the Martin boundary. In this paper we study the KMS states for this flow and obtain a complete description when the Poisson boundary of the random walk is trivial and when \(\Gamma \) is a torsion free non-elementary hyperbolic group. We also construct examples to show that the structure of the KMS states can be more complicated beyond these cases.

Topological flows for hyperbolic groups

AbstractWe show that for every non-elementary hyperbolic group the Bowen–Margulis current associated with a strongly hyperbolic metric forms a unique group-invariant Radon measure class of maximal Hausdorff dimension on the boundary square. Applications include a characterization of roughly similar hyperbolic metrics via mean distortion.

Entropy and drift in word hyperbolic groups

The fundamental inequality of Guivarc’h relates the entropy and the drift of random walks on groups. It is strict if and only if the random walk does not behave like the uniform measure on balls. We prove that, in any nonelementary hyperbolic group which is not virtually free, endowed with a word distance, the fundamental inequality is strict for symmetric measures with finite support, uniformly for measures with a given support. This answers a conjecture of S. Lalley. For admissible measures, this is proved using previous results of Ancona and Blachère–Haïssinsky–Mathieu. For non-admissible measures, this follows from a counting result, interesting in its own right: we show that, in any infinite index subgroup, the number of non-distorted points is exponentially small compared to the growth of balls in the whole group. The uniformity is obtained by studying the behavior of measures that degenerate towards a measure supported on an elementary subgroup.

Property RD and the classification of traces on reduced group C∗-algebras of hyperbolic groups

In this paper, we show that if [Formula: see text] is a non-elementary word hyperbolic group, [Formula: see text] is an element, and the conjugacy class of [Formula: see text] is infinite, then all traces [Formula: see text] vanish on [Formula: see text]. Moreover, we completely classify all traces by showing that traces [Formula: see text] are linear combinations of traces [Formula: see text] given by [Formula: see text] where [Formula: see text] is an element with finite conjugacy class, denoted [Formula: see text]. We demonstrate these two statements by introducing a new method to study traces that uses Sobolev norms and the rapid decay property.

Analyticity of the entropy and the escape rate of random walks in hyperbolic groups

We consider random walks on a non-elementary hyperbolic group endowed with a word distance. To a probability measure on the group are associated two numerical quantities, the rate of escape and the entropy. On the set of admissible probability measures whose support is contained in a given finite set, we show that both quantities depend in an analytic way on the probability measure. Our spectral techniques also give a new proof of the central limit theorem, and imply that the corresponding variance is analytic.

Formal Conjugacy Growth in Acylindrically Hyperbolic Groups

Rivin conjectured that the conjugacy growth series of a hyperbolic group is rational if and only if the group is virtually cyclic. Ciobanu, Hermiller, Holt and Rees proved that the conjugacy growth series of a virtually cyclic group is rational. Here we present the proof confirming the other direction of the conjecture, by showing that the conjugacy growth series of a non-elementary hyperbolic group is transcendental. We also present and prove some variations of Rivin's conjecture for commensurability classes and primitive conjugacy classes. We then explore Rivin's conjecture for finitely generated acylindrically hyperbolic groups and prove a formal language version of it, namely that no set of minimal length conjugacy representatives can be unambiguous context-free.

Hausdorff spectrum of harmonic measure

For every non-elementary hyperbolic group, we show that for every random walk with finitely supported admissible step distribution, the associated entropy equals the drift times the logarithmic volume growth if and only if the corresponding harmonic measure is comparable with Hausdorff measure on the boundary. Moreover, we introduce one parameter family of probability measures which interpolates a Patterson–Sullivan measure and the harmonic measure, and establish a formula of Hausdorff spectrum (multifractal spectrum) of the harmonic measure. We also give some finitary versions of dimensional properties of the harmonic measure.

Differentiating the entropy of random walks on hyperbolic groups

We show that the asymptotic entropy of a random walk on a nonelementary hyperbolic group, with symmetric and bounded increments, is differentiable and we identify its derivative as a correlation. We also prove similar results for the rate of escape.

Verbal subgroups of hyperbolic groups have infinite width

Let $G$ be a non-elementary hyperbolic group. Let $w$ be a group word such that the set $w[G]$ of all its values in $G$ does not coincide with $G$ or 1. We show that the width of verbal subgroup $w(G)=<w[G]>$ is infinite. That is, there is no such $l\in\mathbb Z$ that any $g\in w(G)$ can be represented as a product of $\le l$ values of $w$ and their inverses.

Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups

AbstractGiven a group automorphismϕ: Γ → Γ, one has an action of Γ on itself byϕ-twisted conjugacy, namely,g.x = gxϕ(g-1). The orbits of this action are calledϕ-twisted conjugacy classes. One says that Γ has theR∞-property if there are infinitely manyϕ-twisted conjugacy classes for every automorphismϕof Γ. In this paper we show that SL(n; Z) and its congruence subgroups have the R8-property. Further we show that any (countable) abelian extension of Γ has the R8-property where Γ is a torsion free non-elementary hyperbolic group, or SL(n; Z); Sp(2n; Z) or a principal congruence subgroup of SL(n; Z) or the fundamental group of a complete Riemannian manifold of constant negative curvature.

On Cannon–Thurston maps for relatively hyperbolic groups

Abstract.Baker and Riley gave an inclusion of a free group of rank 3 in a hyperbolic group for which the Cannon–Thurston map is not well-defined. By using their result, we show that every non-elementary hyperbolic group can be included in some hyperbolic group in such a way that the Cannon–Thurston map is not well-defined. In fact we generalize their result to every non-elementary relatively hyperbolic group.

Proper isometric actions of hyperbolic groups on -spaces

AbstractWe show that every non-elementary hyperbolic group $\G $ admits a proper affine isometric action on $L^p(\bd \G \times \bd \G )$, where $\bd \G $ denotes the boundary of $\G $ and $p$ is large enough. Our construction involves a $\G $-invariant measure on $\bd \G \times \bd \G $ analogous to the Bowen–Margulis measure from the ${\rm CAT}(-1)$ setting, as well as a geometric, Busemann-type cocycle. We also deduce that $\G $ admits a proper affine isometric action on the first $\ell ^p$-cohomology group $H^1_{(p)}(\G )$ for large enough $p$.

Unique Cartan decomposition for II 1 factors arising from arbitrary actions of hyperbolic groups

Abstract We prove that for any free ergodic probability measure preserving action Γ → ( X , μ ) $\Gamma \rightarrow (X,\mu )$ of a non-elementary hyperbolic group, or a lattice in a rank one simple Lie group, the associated group measure space II 1 factor L ∞ ( X ) ⋊ Γ ${\mathord {\text{\rm L}}^\infty (X) \rtimes \Gamma }$ has L ∞ ( X ) ${\mathord {\text{\rm L}}^\infty (X)}$ as its unique Cartan subalgebra, up to unitary conjugacy.

DEFINABLE SETS IN A HYPERBOLIC GROUP

We give a description of definable sets P = (p1,…,pm) in a free non-abelian group F and in a torsion-free non-elementary hyperbolic group G. As a corollary we show that proper non-cyclic subgroups of F and G are not definable. This answers Malcev's question posed in 1965 for F.

Quasimorphisms, random walks, and transient subsets in countable groups

We study the interrelations between the theory of quasimorphisms and the theory of random walks on groups, and establish the following transience criterion for subsets of groups: if a subset of a countable group has bounded images under any three linearly independent homogeneous quasimorphisms on the group, the this subset is transient for all nondegenerate random walks on the group. From this it follows, by results of M. Bestvina, K. Fujiwara, J. Birman, W. Menasco, and others, that, in a certain sense, generic elements in the mapping class groups of surfaces are pseudo-Anosov, generic braids in Artin’s braid groups represent prime links and knots, generic elements in the commutant of every nonelementary hyperbolic group have large stable commutator length, etc. Bibliography: 20 titles.

Periodic quotients of hyperbolic and large groups

Let G be either a non-elementary (word) hyperbolic group or a large group (both in the sense of Gromov). In this article we describe several approaches for constructing continuous families of periodic quotients of G with various properties. The first three methods work for any non-elementary hyperbolic group, producing three different continua of periodic quotients of G. They are based on the results and techniques, that were developed by Ivanov and Olshanskii in order to show that there exists an integer n such that G/Gn is an infinite group of exponent n. The fourth approach starts with a large group G and produces a continuum of pairwise non-isomorphic periodic residually finite quotients. Speaking of a particular application, we use each of these methods to give a positive answer to a question of Wiegold from the Kourovka Notebook.

Infinite groups with fixed point properties

We construct finitely generated groups with strong fixed point properties. Let $\mathcal{X}_{ac}$ be the class of Hausdorff spaces of finite covering dimension which are mod-$p$ acyclic for at least one prime $p$. We produce the first examples of infinite finitely generated groups $Q$ with the property that for any action of $Q$ on any $X\in \mathcal{X}_{ac}$, there is a global fixed point. Moreover, $Q$ may be chosen to be simple and to have Kazhdan's property (T). We construct a finitely presented infinite group $P$ that admits no non-trivial action by diffeomorphisms on any smooth manifold in $\mathcal{X}_{ac}$. In building $Q$, we exhibit new families of hyperbolic groups: for each $n\geq 1$ and each prime $p$, we construct a non-elementary hyperbolic group $G_{n,p}$ which has a generating set of size $n+2$, any proper subset of which generates a finite $p$-group.