Related papers: On the commensurability of hyperbolic Coxeter grou…

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 establish some geometric constraints on compact Coxeter polytopes in hyperbolic spaces and show that these constraints can be a very useful tool for the classification problem of reflective anisotropic Lorentzian lattices and cocompact…

A model for a finite group is a set of linear characters of subgroups that can be induced to obtain every irreducible character exactly once. A perfect model for a finite Coxeter group is a model in which the relevant subgroups are the…

Hierarchically hyperbolic spaces (HHSs) are a large class of spaces that provide a unified framework for studying the mapping class group, right-angled Artin and Coxeter groups, and many 3--manifold groups. We investigate strongly…

We determine the maximal hyperbolic reflection groups associated to the quadratic forms $-3x_0^2 + x_1^2 + ... + x_n^2$, $n \ge 2$, and present the Coxeter schemes of their fundamental polyhedra. These groups exist in dimensions up to 13,…

For every dimension d, there is an infinite family of convex co-compact reflection groups of isometries of hyperbolic d-space --- the superideal (simplicial and cubical) reflection groups --- with the property that a random group at any…

In this paper, we obtain a complete classification of 331 finite-volume hyperbolic Coxeter 4-dimensional polytopes with 7 facets.

For right-angled Coxeter groups $W_{\Gamma}$, we obtain a condition on $\Gamma$ that is necessary and sufficient to ensure that $W_{\Gamma}$ is thick and thus not relatively hyperbolic. We show that Coxeter groups which are not thick all…

Assuming that every hyperbolic group is residually finite, we prove the congruence subgroup property for mapping class groups of hyperbolic surfaces of finite type. Under the same assumption, it follows that profinitely equivalent…

In 1974, Kaplinskaja classified all simplicial straight hyperbolic Coxeter prisms. In this paper, we determine precisely which of these prisms are properly quasi-arithmetic or arithmetic. We also present some observations regarding…

After classifying 3-dimensional hyperbolic Coxeter pyramids by means of elementary plane geometry, we calculate growth functions of corresponding Coxeter groups by using Steinberg formula and conclude that growth rates of them are always…

A generic finite presentation defines a word hyperbolic group whose boundary is homeomorphic to the Menger curve. In this article, we produce the first known examples of non-hyperbolic $CAT(0)$ groups whose visual boundary is homeomorphic…

Polypolyhedra are edge-transitive compounds of polyhedra. In this paper we use group theory to determine the number of distinct polypolyhedra whose symmetry group is any given finite irreducible Coxeter group. We apply this result in order…

A group is properly 3-realizable if it is the fundamental group of a compact polyhedron whose universal covering is proper homotopically equivalent to some 3-manifold. We prove that when such a group is also quasi-simply filtered then it…

In this paper, for each finite group $G$, we construct explicitly a non-compact complete finite-volume arithmetic hyperbolic $4$-manifold $M$ such that $\mathrm{Isom}\,M \cong G$, or $\mathrm{Isom}^{+}\,M \cong G$. In order to do so, we use…

In this paper, we obtain the complete classification for compact hyperbolic Coxeter four-dimensional polytopes with eight facets.

We construct a limit aperiodic coloring of hyperbolic groups. Also we construct limit strongly aperiodic strictly balanced tilings of the Davis complex for all Coxeter groups.

We study arithmetic properties of the growth rates of cofinite 3-dimensional hyperbolic Coxeter groups whose dihedral angles are of the form $\frac{\pi}{m}$ for $m=2,3,4,5,6$ and show that the growth rates are always Perron numbers.

This is a survey of some aspects of the large-scale geometry of right-angled Coxeter groups. The emphasis is on recent results on their negative curvature properties, boundaries, and their quasi-isometry and commensurability classification.

Beside simplices, $n$-cubes form an important class of simple polyhedra. Unlike hyperbolic Coxeter simplices, hyperbolic Coxeter $n$-cubes are not classified. We show that there is no hyperbolic Coxeter $n$-cube for $n\geq~6$, and provide a…