Hyperbolic Groups Research Articles

Automorphisms of groups and a higher rank JSJ decomposition I: RAAGs and a higher rank Makanin-Razborov diagram

The JSJ decomposition encodes the automorphisms and the virtually cyclic splittings of a hyperbolic group. For general finitely presented groups, the JSJ decomposition encodes only their splittings. In this sequence of papers we study the automorphisms of a hierarchically hyperbolic group that satisfies some weak acylindricity conditions. To study these automorphisms we construct an object that can be viewed as a higher rank JSJ decomposition. In the first paper we demonstrate our construction in the case of a right angled Artin group. For studying automorphisms of a general HHG we construct what we view as a higher rank Makanin-Razborov diagram, which is the first step in the construction of the higher rank JSJ.

Talagrand’s influence inequality revisited

Let $\mathscr{C}_n=\{-1,1\}^n$ be the discrete hypercube equipped with the uniform probability measure $\sigma_n$. Talagrand's influence inequality (1994) asserts that there exists $C\in(0,\infty)$ such that for every $n\in\mathbb{N}$, every function $f:\mathscr{C}_n\to\mathbb{C}$ satisfies $$\mathrm{Var}_{\sigma_n}(f) \leq C \sum_{i=1}^n \frac{\|\partial_if\|_{L_2(\sigma_n)}^2}{1+\log\big(\|\partial_if\|_{L_2(\sigma_n)}/\|\partial_i f\|_{L_1(\sigma_n)}\big)}.$$ In this work, we undertake a systematic investigation of this and related inequalities via harmonic analytic and stochastic techniques and derive applications to metric embeddings. We prove that Talagrand's inequality extends, up to an additional doubly logarithmic factor, to Banach space-valued functions under the necessary assumption that the target space has Rademacher type 2 and that this doubly logarithmic term can be omitted if the target space admits an equivalent 2-uniformly smooth norm. These are the first vector-valued extensions of Talagrand's influence inequality. We also obtain a joint strengthening of results of Bakry-Meyer (1982) and Naor-Schechtman (2002) on the action of negative powers of the hypercube Laplacian on functions $f:\mathscr{C}_n\to E$, whose target space $E$ has nontrivial Rademacher type via a new vector-valued version of Meyer's multiplier theorem (1984). Inspired by Talagrand's influence inequality, we introduce a new metric invariant called Talagrand type and estimate it for Banach spaces with prescribed Rademacher or martingale type, Gromov hyperbolic groups and simply connected Riemannian manifolds of pinched negative curvature. Finally, we prove that Talagrand type is an obstruction to the bi-Lipschitz embeddability of nonlinear quotients of the hypercube $\mathscr{C}_n$, thus deriving new nonembeddability results for these finite metrics.

Rips construction without unique product

Given a finitely presented group $Q,$ we produce a short exact sequence $1\to N \hookrightarrow G \twoheadrightarrow Q \to 1$ such that $G$ is a torsion-free Gromov hyperbolic group without the unique product property and $N$ is without the unique product property and has Kazhdan's Property (T). Varying $Q,$ we show a wide diversity of concrete examples of Gromov hyperbolic groups without the unique product property. As an application, we obtain Tarski monster groups without the unique product property.

Manifolds without real projective or flat conformal structures

In any dimension at least five we construct examples of closed smooth manifolds with the following properties: (1) they have neither real projective nor flat conformal structures; (2) their fundamental group is a non-elementary Gromov hyperbolic group. These examples are obtained via relative strict hyperbolization.

Strongly aperiodic SFTs on hyperbolic groups: where to find them and why we love them

Strongly aperiodic SFTs on hyperbolic groups: where to find them and why we love them

Relatively hyperbolic groups with strongly shortcut parabolics are strongly shortcut

AbstractWe show that a group that is hyperbolic relative to strongly shortcut groups is itself strongly shortcut, thus obtaining new examples of strongly shortcut groups. The proof relies on a result of independent interest: we show that every relatively hyperbolic group acts properly and cocompactly on a graph in which the parabolic subgroups act properly and cocompactly on convex subgraphs.

Nielsen equivalence in triangle groups

We show that any non-standard generating pair of a hyperbolic triangle group is represented by a special almost orbifold covering with a good marking.

Classifiability of crossed products by nonamenable groups

Abstract We show that all amenable, minimal actions of a large class of nonamenable countable groups on compact metric spaces have dynamical comparison. This class includes all nonamenable hyperbolic groups, many HNN-extensions, nonamenable Baumslag–Solitar groups, a large class of amalgamated free products, lattices in many Lie groups, A ~ 2 {\widetilde{A}_{2}} -groups, as well as direct products of the above with arbitrary countable groups. As a consequence, crossed products by amenable, minimal and topologically free actions of such groups on compact metric spaces are Kirchberg algebras in the UCT class, and are therefore classified by K-theory.

Expanders are counterexamples to the $\ell^p$ coarse Baum–Connes conjecture

We consider an \ell^p coarse Baum–Connes assembly map for 1 < p < \infty , and show that it is not surjective for expanders arising from residually finite hyperbolic groups.

On canonical splittings of relatively hyperbolic groups

A JSJ decomposition of a group is a splitting that allows one to classify all possible splittings of the group over a certain family of edge groups. Although JSJ decompositions are not unique in general, Guirardel–Levitt have constructed a canonical JSJ decomposition, the tree of cylinders, which classifies splittings of relatively hyperbolic groups over elementary subgroups. In this paper, we give a new topological construction of the Guirardel–Levitt tree of cylinders, and we show that this tree depends only on the homeomorphism type of the Bowditch boundary. Furthermore, the tree of cylinders admits a natural action by the group of homeomorphisms of the boundary. In particular, the quasi-isometry group of (G, ℙ) acts naturally on the tree of cylinders.

The Martin boundary of an extension by a hyperbolic group

The Martin boundary of an extension by a hyperbolic group

Unbounded domains in hierarchically hyperbolic groups

We investigate unbounded domains in hierarchically hyperbolic groups and obtain constraints on the possible hierarchical structures. Using these insights, we characterise the structures of virtually abelian HHGs and show that the class of HHGs is not closed under finite extensions. This provides a strong negative answer to the question of whether being an HHG is invariant under quasi-isometries. Along the way, we show that infinite torsion groups are not HHGs. By ruling out pathological behaviours, we are able to give simpler, direct proofs of the rank-rigidity and omnibus subgroup theorems for HHGs. This involves extending our techniques so that they apply to all subgroups of HHGs.

Complexity of word problems for HNN-extensions

The computational complexity of the word problem in HNN-extension of groups is studied. HNN-extension is a fundamental construction in combinatorial group theory. It is shown that the word problem for an ascending HNN-extension of a group H is logspace reducible to the so-called compressed word problem for H. The main result of the paper states that the word problem for an HNN-extension of a hyperbolic group H with cyclic associated subgroups can be solved in polynomial time. This result can be easily extended to fundamental groups of graphs of groups with hyperbolic vertex groups and cyclic edge groups.

Non-colorable hierarchically hyperbolic groups

We exhibit a hierarchically hyperbolic group for which no hierarchically hyperbolic structure is colorable, answering an (implicit) question of Durham–Minsky–Sisto.

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.

A report on an ergodic dichotomy

AbstractWe establish (some directions of) a Ledrappier correspondence between Hölder cocycles, Patterson–Sullivan measures, etc for word-hyperbolic groups with metric-Anosov Mineyev flow. We then study Patterson–Sullivan measures for $\vartheta $ -Anosov representations over a local field and show that these are parameterized by the $\vartheta $ -critical hypersurface of the representation. We use these Patterson–Sullivan measures to establish a dichotomy concerning directions in the interior of the $\vartheta $ -limit cone of the representation in question: if ${\mathsf {u}}$ is such a half-line, then the subset of ${\mathsf {u}}$ -conical limit points has either total mass if $|\vartheta |\leq 2$ or zero mass if $|\vartheta |\geq 4.$ The case $|\vartheta |=3$ remains unsettled.

Generating hyperbolic isometry groups by elementary matrices

We consider three families of groups: the Bianchi groups where is the ring of integers of an imaginary, quadratic field; the groups where is a -order of a definite, rational quaternion algebra with an orthogonal involution; and the groups where is an order of a definite, rational quaternion algebra. We show that such groups are generated by elementary matrices if and only if is semi-Euclidean (or -semi-Euclidean), which is a generalization of the usual notion of a Euclidean ring. The proofs are surprisingly simple and proceed by considering fundamental domains of Kleinian groups.

Characterizing candidates for Cannon's conjecture from geometric measure theory

AbstractWe show that recent work of Song implies that torsion‐free hyperbolic groups with Gromov boundary arerealized as fundamental groups of closed 3‐manifolds of constant negative curvature if and only if the solution to an associated spherical Plateau problem for group homology is isometric to such a 3‐manifold, and suggest some related questions.

On Subgroups Finite Index in Complex Hyperbolic Lattice Triangle Groups

We study several explicit finite index subgroups in the known complex hyperbolic lattice triangle groups, and show some of them are neat, some of them have positive first Betti number, some of them have a homomorphisms onto a non-Abelian free group. For some lattice triangle groups, we determine the minimal index of a neat subgroup. Finally, we answer a question raised by Stover and describe an infinite tower of neat ball quotients all with a single cusp.

Finitely generated groups acting uniformly properly on hyperbolic space

We construct an uncountable sequence of groups acting uniformly properly on hyperbolic spaces. We show that only countably many of these groups can be virtually torsion-free. This gives new examples of groups acting uniformly properly on hyperbolic spaces that are not virtually torsion-free and cannot be subgroups of hyperbolic groups.