Hyperbolic Groups Research Articles

Complex hyperbolic Kleinian groups of large critical exponents

Complex hyperbolic Kleinian groups of large critical exponents

Residually finite non-linear hyperbolic groups

We exhibit the first examples of residually finite non-linear Gromov hyperbolic groups. Our examples are constructed as amalgamated products of torsion-free cocompact lattices in the rank 1 Lie group \mathsf{Sp}(d,1) , d \geq2 , along maximal cyclic subgroups.

Finitely Presented Kernels of Homomorphisms From Hyperbolic Groups Onto Free Abelian Groups

Finitely Presented Kernels of Homomorphisms From Hyperbolic Groups Onto Free Abelian Groups

The algebraic structure of hyperbolic graph braid groups

The algebraic structure of hyperbolic graph braid groups

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.

Divisible convex sets with properly embedded cones

AbstractIn this article we construct many examples of properly convex irreducible domains divided by Zariski dense relatively hyperbolic groups in every dimension at least 3. This answers a question of Benoist. Relative hyperbolicity and non-strict convexity are captured by a family of properly embedded cones (convex hulls of points and ellipsoids) in the domain. Our construction is most flexible in dimension 3 where we give a purely topological criterion for the existence of a large deformation space of geometrically controlled convex projective structures with totally geodesic boundary on a compact 3-manifold.

Soluble quotients of triangle groups

This paper helps explain the prevalence of soluble groups among the automorphism groups of regular maps (at least for ‘small’ genus), by showing that every non-perfect hyperbolic ordinary triangle group Δ+(p,q,r)=〈x,y|xp=yq=(xy)r=1〉 has a smooth finite soluble quotient of derived length c for some c≤3, and infinitely many such quotients of derived length d for every d>c.

On the hyperbolic group and subordinated integrals as operators on sequence Banach spaces

AbstractWe show that the composition hyperbolic group in the unit disc, once transferred to act on sequence spaces, is bounded on $$\ell^p$$ ℓ p if and only if $${p=2}$$ p = 2 . We introduce some integral operators subordinated to that group which are natural generalizations of classical operators on sequences. For the description of such operators, we use some combinatorial identities which look interesting in their own.

The rates of growth in an acylindrically hyperbolic group

Let G be an acylindrically hyperbolic group on a \delta -hyperbolic space X . Assume there exists M such that for any finite generating set S of G , the set S^{M} contains a hyperbolic element on X . Suppose that G is equationally Noetherian. Then we show the set of the growth rates of G is well ordered. The conclusion was known for hyperbolic groups, and this is a generalization. Our result applies to all lattices in simple Lie groups of rank 1, and more generally, relatively hyperbolic groups under some assumption. It also applies to the fundamental group, of exponential growth, of a closed orientable 3 -manifold except for the case that the manifold has Sol-geometry.

Riesz operators and Lp-boundary representations for hyperbolic groups

We investigate Lp-boundary representations of hyperbolic groups. We prove that such representations are irreducible if and only if the corresponding Riesz operators are injective.

Morse subsets of injective spaces are strongly contracting

We show that a quasi-geodesic in an injective metric space is Morse if and only if it is strongly contracting. Since mapping class groups and, more generally, hierarchically hyperbolic groups act properly and coboundedly on injective metric spaces, we deduce various consequences relating, for example, to growth tightness and genericity of pseudo-Anosovs/Morse elements. Moreover, we show that injective metric spaces have the Morse local-to-global property and that a non-virtually cyclic group acting properly and coboundedly on an injective metric space is acylindrically hyperbolic if and only if contains a Morse ray. We also show that strongly contracting geodesics of a space stay strongly contracting in the injective hull of that space.

Deciding if a hyperbolic group splits over a given quasiconvex subgroup

We present an algorithm which decides whether a given quasiconvex residually finite subgroup H of a hyperbolic group G is associated with a splitting. The methods developed also provide algorithms for computing the number of filtered ends \tilde{e}(G,H) of H in G under certain hypotheses and give a new straightforward algorithm for computing the number of ends e(G,H) of the Schreier graph of H . Our techniques extend those of Barrett via the use of labelled digraphs, the languages of which encode information on the connectivity of \partial G - \Lambda H .

A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups

AbstractWe give a simple combinatorial criterion, in terms of an action on a hyperbolic simplicial complex, for a group to be hierarchically hyperbolic. We apply this to show that quotients of mapping class groups by large powers of Dehn twists are hierarchically hyperbolic (and even relatively hyperbolic in the genus 2 case). In genus at least three, there are no known infinite hyperbolic quotients of mapping class groups. However, using the hierarchically hyperbolic quotients we construct, we show, under a residual finiteness assumption, that mapping class groups have many nonelementary hyperbolic quotients. Using these quotients, we relate questions of Reid and Bridson–Reid–Wilton about finite quotients of mapping class groups to residual finiteness of specific hyperbolic groups.

Reflection length at infinity in hyperbolic reflection groups

Abstract In a discrete group generated by hyperplane reflections in the 𝑛-dimensional hyperbolic space, the reflection length of an element is the minimal number of hyperplane reflections in the group that suffices to factor the element. For a Coxeter group that arises in this way and does not split into a direct product of spherical and affine reflection groups, the reflection length is unbounded. The action of the Coxeter group induces a tessellation of the hyperbolic space. After fixing a fundamental domain, there exists a bijection between the tiles and the group elements. We describe certain points in the visual boundary of the 𝑛-dimensional hyperbolic space for which every neighbourhood contains tiles of every reflection length. To prove this, we show that two disjoint hyperplanes in the 𝑛-dimensional hyperbolic space without common boundary points have a unique common perpendicular.

Hyperfiniteness of boundary actions of relatively hyperbolic groups

We show that if G is a finitely generated group hyperbolic relative to a finite collection of subgroups \mathcal{P} , then the natural action of G on the geodesic boundary of the associated relative Cayley graph induces a hyperfinite equivalence relation. As a corollary of this, we obtain that the natural action of G on its Bowditch boundary \partial (G,\mathcal{P}) also induces a hyperfinite equivalence relation. This strengthens a result of Ozawa obtained for \mathcal{P} consisting of amenable subgroups and uses a recent work of Marquis and Sabok.

Compressed decision problems in hyperbolic groups

We prove that, for any hyperbolic group, the compressed word and the compressed conjugacy problems are solvable in polynomial time. As a consequence, the word problem for the (outer) automorphism group of a hyperbolic group is solvable in polynomial time. We also prove that the compressed simultaneous conjugacy and the compressed centraliser problems are solvable in polynomial time. Finally, we prove that, for any infinite hyperbolic group, the compressed knapsack problem is \mathrm{NP} -complete.

Homeomorphism types of the Bowditch boundaries of infinite-ended relatively hyperbolic groups

Abstract For n ≥ 2 n\geq 2 , let G 1 = A 1 ∗ ⋯ ∗ A n G_{1}=A_{1}\ast\dots\ast A_{n} and G 2 = B 1 ∗ ⋯ ∗ B n G_{2}=B_{1}\ast\dots\ast B_{n} where the A i A_{i} ’s and B i B_{i} ’s are non-elementary relatively hyperbolic groups. Suppose that, for 1 ≤ i ≤ n 1\leq i\leq n , the Bowditch boundary of A i A_{i} is homeomorphic to the Bowditch boundary of B i B_{i} . We show that the Bowditch boundary of G 1 G_{1} is homeomorphic to the Bowditch boundary of G 2 G_{2} . We generalize this result to graphs of relatively hyperbolic groups with finite edge groups. This extends Martin–Świątkowski’s work in the context of relatively hyperbolic groups.

Two-parabolic-generator subgroups of hyperbolic 3-manifold groups

Two-parabolic-generator subgroups of hyperbolic 3-manifold groups

Weak [formula omitted]-structures for some combinatorial group constructions

Bestvina [1] introduced the notion of a (weak) Z-structure and (weak) Z-boundary for a torsion-free group, motivated by the notion of boundary for hyperbolic and CAT(0) groups. Since then, some classes of groups have been shown to admit a (weak) Z-structure (see [5,20,22] for example); in fact, in all cases these groups are semistable at infinity and happen to have a pro-(finitely generated free) fundamental pro-group. The question whether or not every type F group admits such a structure remains open. In [33] it was shown that the property of admitting such a structure is closed under direct products and free products. Our main results are as follows.THEOREM: Let G be a torsion-free and semistable at infinity finitely presented group with a pro-(finitely generated free) fundamental pro-group at each end. If G has a finite graph of groups decomposition in which all the groups involved are of type F and inward tame (in particular, if they all admit a weak Z-structure) then G admits a weak Z-structure.COROLLARY: The class of those 1-ended and semistable at infinity torsion-free finitely presented groups which admit a weak Z-structure and have a pro-(finitely generated free) fundamental pro-group is closed under amalgamated products (resp. HNN-extensions) over finitely generated free groups.On the other hand, given a finitely presented group G and a monomorphism φ:G⟶G, we may consider the ascending HNN-extension G⁎φ=〈G,t;t−1gt=φ(g),g∈G〉. The results in [26] together with the Theorem above yield the following:PROPOSITION: If a finitely presented torsion-free group G is of type F and inward tame, then any (1-ended) ascending HNN-extension G⁎φ admits a weak Z-structure.In the particular case φ∈Aut(G), this ascending HNN-extension corresponds to a semidirect product G⋊φZ, and it has been shown in [18] that if G admits a Z-structure then so does G⋊φZ.

Frattini subgroups of hyperbolic-like groups

We study Frattini subgroups of various generalizations of hyperbolic groups. For any countable group G admitting a general type action on a hyperbolic space S, we show that the induced action of the Frattini subgroup Φ(G) on S has bounded orbits. This implies that Φ(G) is “small” compared to G; in particular, |G:Φ(G)|=∞. In contrast, for any finitely generated non-cyclic group Q with Φ(Q)={1}, we construct an infinite lacunary hyperbolic group L such that L/Φ(L)≅Q; in particular, the Frattini subgroup of an infinite lacunary hyperbolic group can have finite index. As an application, we obtain the first examples of invariably generated, infinite, lacunary hyperbolic groups.