Related papers: Infinitely many hyperbolic Coxeter groups through …

We show that there is no compact hyperbolic Coxeter d-polytope with d+4 facets for d>7. This bound is sharp: examples of such polytopes up to dimension 7 were found by Bugaenko (1984). We also show that in dimension d=7 the polytope with 11…

We construct some cusped finite-volume hyperbolic $n$-manifolds $M_n$ that fiber algebraically in all the dimensions $5\leq n \leq 8$. That is, there is a surjective homomorphism $\pi_1(M_n) \to \mathbb Z$ with finitely generated kernel.…

We exhibit two examples of convex cocompact subgroups of the isometry groups of real hyperbolic spaces with limit set a Pontryagin sphere: one generated by $50$ reflections of $\mathbb{H}^4$, and the other by a rotation of order $21$ and a…

We consider the class of countable groups possessing an action on a finite product of hyperbolic graphs where every infinite order element acts loxodromically. When the graphs are locally finite, we obtain strong structure theorems for the…

In this paper we show how to obtain representations of Coxeter groups acting on H^n to certain classical groups. We determine when the kernel of such a representation is torsion-free and thus the quotient a hyperbolic n-manifold.

We complete the classification of compact hyperbolic Coxeter $d$-polytopes with $d+4$ facets for $d=4$ and $5$. By previous work of Felikson and Tumarkin, the only remaining dimension where new polytopes may arise is $d=6$. We derive a new…

We provide a general construction of convex cocompact hyperbolic reflection groups with three-dimensional limit sets. More precisely, our construction takes as input an arbitrary simplicial complex L of dimension 3 on n vertices, and…

Coxeter groups admit amenable actions on compact spaces. Moreover, they have finite asymptotic dimension.

We show that right-angled Coxeter groups are relatively hyperbolic in the sense defined by Farb, relative to a natural collection of rank-2 parabolic subgroups.

We prove that a finitely generated, right-angled, hyperbolic Coxeter group can be quasiisometrically embedded into the product of n binary trees, where n is the chromatic number of the group. As application we obtain certain strongly…

Let G be a group admitting a non-elementary acylindrical action on a Gromov hyperbolic space (for example, a non-elementary relatively hyperbolic group, or the mapping class group of a closed hyperbolic surface, or Out(F_n) for n>1). We…

In this paper we study $\times_0$-products of Lann\'er diagrams. We prove that every $\times_0$-product of at least four Lann\'er diagrams with at least one diagram of order $\ge 3$ is superhyperbolic. As a corollary, we obtain that known…

We show that all groups in a very large class of Coxeter groups are locally quasiconvex and have uniform membership problem solvable in quadratic time. If a group in the class satisfies a further hypothesis it is subgroup separable and…

For $\Gamma$ a relatively hyperbolic group, we construct a model for the universal space among $\Gamma$-spaces with isotropy on the family VC of virtually cyclic subgroups of $\Gamma$. We provide a recipe for identifying the maximal…

We prove that every non-elementary hyperbolic group $G$ acts with maximal growth on some set $X$ such that every orbit of any element $g \in G$ is finite. As a side-product of our approach we prove that if $G$ is non-elementary hyperbolic,…

Let X be an arbitrary hyperbolic geodesic metric space and let G be a countable non-elementary weakly acylindrical group of isometries of X. We show that the second bounded cohomology group of G with real coefficients or with coefficients…

Suppose that a finitely generated group $G$ is hyperbolic relative to a collection of subgroups $\{H_1, ..., H_m\} $. We prove that if each of the subgroups $H_1, ..., H_m$ has finite asymptotic dimension, then asymptotic dimension of $G$…

We construct infinite series of non-simple ideal hyperbolic Coxeter 4-polytopes whose growth rates are Perron numbers. This infinite series is the first example of such a non-compact infinite polytopal series.

In contrast to the fact that there are only finitely many maximal arithmetic reflection groups acting on the hyperbolic space $\mathbb{H}^n$, $n\geq 2$, we show that: (a) one can produce infinitely many maximal quasi-arithmetic reflection…

The isometry group of a compact n-dimensional hyperbolic manifold is known to be finite. We show that for every n > 2, every finite group is realized as the full isometry group of some compact hyperbolic n-manifold. The cases n = 2 and n =…