Hyperbolic Groups Research Articles

Propagating quasiconvexity from fibers

Let \(1 \to K \to G\,\mathop \to \limits^\pi \,Q\) be an exact sequence of hyperbolic groups. Let Q1 < Q be a quasiconvex subgroup and let G1 = π−1(Q1). Under relatively mild conditions (e.g. if K is a closed surface group or a free group and Q is convex cocompact), we show that infinite index quasiconvex subgroups of G1 are quasiconvex in G. Related results are proven for metric bundles, developable complexes of groups and graphs of groups.

Systolic length of triangular modular curves

We present a method for computing upper bounds on the systolic length of certain Riemann surfaces uniformized by congruence subgroups of hyperbolic triangle groups, admitting congruence Hurwitz curves as a special case. The uniformizing group is realized as a Fuchsian group and a convenient finite generating set is computed. The upper bound is derived from the traces of the generators. Some explicit computations, including ones for non-arithmetic surfaces, are given. We apply a result of Cosac and Dória to show that the systolic length grows logarithmically with respect to the genus.

On the topological complexity of toral relatively hyperbolic groups

We prove that the topological complexity TC ⁡ ( π ) \operatorname {TC}(\pi ) equals c d ( π × π ) cd(\pi \times \pi ) for certain toral relatively hyperbolic groups π \pi .

Cross ratios and cubulations of hyperbolic groups

Many geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichmüller space, Hitchin representations and geodesic currents. We add to this picture by studying cocompact cubulations of arbitrary Gromov hyperbolic groups G. Under weak assumptions, we show that the space of cubulations of G naturally injects into the space of G-invariant cross ratios on the Gromov boundary partial _{infty }G. A consequence of our results is that essential, hyperplane-essential, cocompact cubulations of hyperbolic groups are length-spectrum rigid, i.e. they are fully determined by their length function. This is the optimal length-spectrum rigidity result for cubulations of hyperbolic groups, as we demonstrate with some examples. In the hyperbolic setting, this constitutes a strong improvement on our previous work [4]. Along the way, we describe the relationship between the Roller boundary of a mathrm{CAT(0)} cube complex, its Gromov boundary and—in the non-hyperbolic case—the contracting boundary of Charney and Sultan. All our results hold for cube complexes with variable edge lengths.

Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

We describe homomorphisms $$\varphi :H\rightarrow G$$ for which G is acylindrically hyperbolic and H is a topological group which is either completely metrizable or locally countably compact Hausdorff. It is shown that, in a certain sense, either the image of $$\varphi $$ is small or $$\varphi $$ is almost continuous. We also describe homomorphisms from the Hawaiian earring group to G as above. We prove a more precise result for homomorphisms $$\varphi :H\rightarrow {\text {Mod}}(\Sigma )$$ , where H is as above and $${\text {Mod}}(\Sigma )$$ is the mapping class group of a connected compact surface $$\Sigma $$ . In this case there exists an open normal subgroup $$V\leqslant H$$ such that $$\varphi (V)$$ is finite. We also prove the analogous statement for homomorphisms $$\varphi :H\rightarrow {\text {Out}}(G)$$ , where G is a one-ended hyperbolic group. Some automatic continuity results for relatively hyperbolic groups and fundamental groups of graphs of groups are also deduced. As a by-product, we prove that the Hawaiian earring group is acylindrically hyperbolic, but does not admit any universal acylindrical action on a hyperbolic space.

Hyperbolic 3-manifold groups are subgroup into conjugacy separable

A group G is subgroup conjugacy distinguished (resp. subgroup into conjugacy separable) if whenever an element y (resp. a finitely generated subgroup K) is not conjugate to an element (to a subgroup) of a finitely generated subgroup H of G there exists a finite quotient G/N of G where yN (resp. KN/N) is not conjugate to an element (to a subgroup) of HN/N. We prove that the fundamental group of a hyperbolic 3-manifold is subgroup conjugacy distinguished and subgroup into conjugacy separable.

Cannon–Thurston maps for {{,textrm{CAT},}}(0) groups with isolated flats

Mahan Mitra (Mj) proved Cannon–Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group (Mitra in Topology, 37(3):527–538, 1998). We prove that Cannon–Thurston maps do not exist for infinite normal hyperbolic subgroups of non-hyperbolic {{,textrm{CAT},}}(0) groups with isolated flats with respect to the visual boundaries. We also show Cannon–Thurston maps do not exist for infinite infinite-index normal {{,textrm{CAT},}}(0) subgroups with isolated flats in non-hyperbolic {{,textrm{CAT},}}(0) groups with isolated flats. We obtain a structure theorem for the normal subgroups in these settings and show that outer automorphism groups of hyperbolic groups have no purely atoroidal mathbb {Z}^2 subgroups.

Projective Anosov representations, convex cocompact actions, and rigidity

In this paper we show that many projective Anosov representations act convex cocompactly on some properly convex domain in real projective space. In particular, if a non-elementary word hyperbolic group is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then any projective Anosov representation of that group acts convex cocompactly on some properly convex domain in real projective space. We also show that if a projective Anosov representation preserves a properly convex domain, then it acts convex cocompactly on some (possibly different) properly convex domain. We then give three applications. First, we show that Anosov representations into general semisimple Lie groups can be defined in terms of the existence of a convex cocompact action on a properly convex domain in some real projective space (which depends on the semisimple Lie group and parabolic subgroup). Next, we prove a rigidity result involving the Hilbert entropy of a projective Anosov representation. Finally, we prove a rigidity result which shows that the image of the boundary map associated to a projective Anosov representation is rarely a $C^2$ submanifold of projective space. This final rigidity result also applies to Hitchin representations.

A spectral sequence for Dehn fillings

We study how the cohomology of a type $F_\infty$ relatively hyperbolic group pair $(G,\mathcal{P})$ changes under Dehn fillings (i.e. quotients of group pairs). For sufficiently long Dehn fillings where the quotient pair $(\bar{G},\bar{\mathcal{P}})$ is of type $F_\infty$, we show that there is a spectral sequence relating the cohomology groups $H^i(G,\mathcal{P};\mathbb{Z} G)$ and $H^i\left(\bar{G},\bar{\mathcal{P}};\mathbb{Z}\bar{G}\right)$. As a consequence, we show that essential cohomological dimension does not increase under these Dehn fillings.

Short, Highly Imprimitive Words Yield Hyperbolic One-Relator Groups

We give experimental support for a conjecture of Louder and Wilton saying that words of imprimitivity rank greater than two yield hyperbolic one-relator groups.

Redundancy of Triangle Groups in Spherical CR Representations

Falbel, Koseleff and Rouillier computed a large number of boundary unipotent CR representations of fundamental groups of non compact three-manifolds. Those representations are not always discrete. By experimentally computing their limit set, one can determine that those with fractal limit sets are discrete. Many of those discrete representations can be related to complex hyperbolic triangle groups. By exact computations, we verify the existence of those triangle representations, which have boundary unipotent holonomy. We also show that many representations are redundant: for n fixed, all the representations encountered are conjugate and only one among them is uniformizable.

Conformal dimension of hyperbolic groups that split over elementary subgroups

We study the (Ahlfors regular) conformal dimension of the boundary at infinity of Gromov hyperbolic groups which split over elementary subgroups. If such a group is not virtually free, we show that the conformal dimension is equal to the maximal value of the conformal dimension of the vertex groups, or 1, whichever is greater, and we characterise when the conformal dimension is attained. As a consequence, we are able to characterise which Gromov hyperbolic groups (without 2-torsion) have conformal dimension 1, answering a question of Bonk and Kleiner.

The relative exponential growth rate of subgroups of acylindrically hyperbolic groups

Abstract We introduce a new invariant of finitely generated groups, the ambiguity function, and we prove that every finitely generated acylindrically hyperbolic group has a linearly bounded ambiguity function. We use this result to prove that the relative exponential growth rate lim n → ∞ ⁡ | B H X ⁢ ( n ) | n \lim_{n\to\infty}\sqrt[n]{\lvert\vphantom{1_{1}}{B^{X}_{H}(n)}\rvert} of a subgroup 𝐻 of a finitely generated acylindrically hyperbolic group 𝐺 exists with respect to every finite generating set 𝑋 of 𝐺 if 𝐻 contains a loxodromic element of 𝐺. Further, we prove that the relative exponential growth rate of every finitely generated subgroup 𝐻 of a right-angled Artin group A Γ A_{\Gamma} exists with respect to every finite generating set of A Γ A_{\Gamma} .

The complexity of solution sets to equations in hyperbolic groups

We show that the full set of solutions to systems of equations and inequations in a hyperbolic group, as shortlex geodesic words (or any regular set of quasigeodesic normal forms), is an EDT0L language whose specification can be computed in NSPACE(n2 log n) for the torsion-free case and NSPACE(n4 log n) in the torsion case. Furthermore, in the presence of effective quasi-isometrically embeddable rational constraints, we show that the full set of solutions to systems of equations in a hyperbolic group remains EDT0L. Our work combines the geometric results of Rips, Sela, Dahmani and Guirardel on the decidability of the existential theory of hyperbolic groups with the work of computer scientists including Plandowski, Jeż, Diekert and others on PSPACE algorithms to solve equations in free monoids and groups using compression, and involves an intricate language-theoretic analysis.

A note on hyperbolically embedded subgroups

Let G be a group and H a subgroup of G. The notion of H being hyperbolically embedded in G was introduced by Dahmani, Guiraldel, and Osin. This note introduces an equivalent definition of hyperbolic embedded subgroup based on Bowditch’s approach to relatively hyperbolic groups in terms of fine graphs.

Strongly aperiodic subshifts of finite type on hyperbolic groups

AbstractWe prove that a hyperbolic group admits a strongly aperiodic subshift of finite type if and only if it has at most one end.

Random walks and quasi‐convexity in acylindrically hyperbolic groups

Arzhantseva proved that every infinite-index quasi-convex subgroup H of a torsion-free hyperbolic group G is a free factor in a larger quasi-convex subgroup of G. We give a probabilistic generalization of this result. That is, we show that when R is a subgroup generated by independent random walks in G, then ⟨ H , R ⟩ ≅ H * R with probability going to one as the lengths of the random walks go to infinity and this subgroup is quasi-convex in G. Moreover, our results hold for a large class of groups acting on hyperbolic metric spaces and subgroups with quasi-convex orbits. In particular, when G is the mapping class group of a surface and H is a convex cocompact subgroup we show that ⟨ H , R ⟩ is convex cocompact and isomorphic to H * R .

Deforming cubulations of hyperbolic groups

We describe a procedure to deform cubulations of hyperbolic groups by "bending hyperplanes". Our construction is inspired by related constructions like Thurston's Mickey Mouse example, walls in fibred hyperbolic $3$-manifolds and free-by-$\mathbb Z$ groups, and Hsu-Wise turns. As an application, we show that every cocompactly cubulated Gromov-hyperbolic group admits a proper, cocompact, essential action on a ${\rm CAT}(0)$ cube complex with a single orbit of hyperplanes. This answers (in the negative) a question of Wise, who proved the result in the case of free groups. We also study those cubulations of a general group $G$ that are not susceptible to trivial deformations. We name these "bald cubulations" and observe that every cocompactly cubulated group admits at least one bald cubulation. We then apply the hyperplane-bending construction to prove that every cocompactly cubulated hyperbolic group $G$ admits infinitely many bald cubulations, provided $G$ is not a virtually free group with ${\rm Out}(G)$ finite. By contrast, we show that the Burger-Mozes examples each admit a unique bald cubulation.

The visual boundary of hyperbolic free-by-cyclic groups

Let ϕ be an atoroidal outer automorphism of the free group Fn. We study the Gromov boundary of the hyperbolic group Gϕ = Fn ⋊ϕ ℤ. Using the Cannon-Thurston map, we explicitly describe a family of embeddings of the complete bipartite graph K3,3 into the boundary of the free-by-cyclic group. To do so, we define the directional Whitehead graph and use it to relate the topology of the boundary to the structure of the Rips Machine associated to a fully irreducible outer automorphism of the free group. In particular, we prove that an indecomposable Fn-tree is Levitt type if and only if one of its directional Whitehead graphs contains more than one edge. As an application, we obtain a new proof of Kapovich-Kleiner’s theorem [KK00] that ∂Gϕ is homeomorphic to the Menger curve if the automorphism is atoroidal and fully irreducible.

Towers and elementary embeddings in toral relatively hyperbolic groups

In a remarkable series of papers, Zlil Sela classified the first-order theories of free groups and torsion-free hyperbolic groups using geometric structures he called towers. It was later proved by Chloé Perin that if [Formula: see text] is an elementarily embedded subgroup (or elementary submodel) of a torsion-free hyperbolic group [Formula: see text], then [Formula: see text] is a tower over [Formula: see text]. We prove a generalization of Perin’s result to toral relatively hyperbolic groups using JSJ and shortening techniques.