Related papers: Coxeter decompositions of hyperbolic simplices
In this paper we introduce the galaxy of Coxeter groups -- an infinite dimensional, locally finite, ranked simplicial complex which captures isomorphisms between Coxeter systems. In doing so, we would like to suggest a new framework to…
Let $C(L)$ be the right-angled Coxeter group defined by an abstract triangulation $L$ of $\mathbb{S}^2$. We show that $C(L)$ is isomorphic to a hyperbolic right-angled reflection group if and only if $L$ can be realized as an acute…
A convex polyhedron, that is, a compact convex subset of $\mathbb{R}^3$ which is the intersection of finitely many closed half-spaces, can be rectified by taking the convex hull of the midpoints of the edges of the polyhedron. We derive…
In this paper we study the problem of hyperball (hypersphere) packings in $3$-dimensional hyperbolic space. We introduce a new definition of the non-compact saturated ball packings and describe to each saturated hyperball packing, a new…
In our previous paper math.QA/0409261, we defined a deformation of the group algebra of the group of even elements of a Coxeter group W, and showed that it is flat for all values of parameters if and only if all the rank 3 parabolic…
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…
A hyperbolic reflection group is a discrete group generated by reflections in the faces of an $n$-dimensional hyperbolic polyhedron. This survey article is dedicated to the study of arithmetic hyperbolic reflection groups with an emphasis…
By a recent result obtained by R. Howlett and the author considerable progress has been made towards a complete solution of the isomorphism problem for Coxeter groups. In this paper we give a survey on the isomorphism problem and explain in…
Coxeter polynomials are important homological invariants that are defined for a large class of finite-dimensional algebras. It is of particular interest to develop methods to compute these polynomials. We define the notion of insertion of a…
A hyperbolic lattice is called \textit{$(1{,}2)$-reflective} if its automorphism group is generated by $1$- and $2$-reflections up to finite index. In this paper we prove that the fundamental polyhedron of a $\mathbb{Q}$-arithmetic…
Let $M$ be a volume finite non-compact complete hyperbolic $n$-manifold with totally geodesic boundary. We show that there exists a polyhedral decomposition of $M$ such that each cell is either an ideal polyhedron or a partially truncated…
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…
A representation of a finitely generated group into the projective general linear group is called convex co-compact if it has finite kernel and its image acts convex co-compactly on a properly convex domain in real projective space. We…
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…
The associahedron is a convex polytope whose face poset is based on nonintersecting diagonals of a convex polygon. In this paper, given an arbitrary simple polygon P, we construct a polytopal complex analogous to the associahedron based on…
In this paper we will consider the 2-fold symmetric complex hyperbolic triangle groups generated by three complex reflections through angle 2pi/p with p no smaller than 2. We will mainly concentrate on the groups where some elements are…
We give a cohomological interpretation of the geodesic simplices of the pseudo-hyperbolic space of signature $(p,q)$ and formulate a necessary and sufficient condition for such a simplex to have finite volume. As a corollary, we obtain that…
Tetrahedron equation is a three dimensional analogue of the Yang-Baxter equation. It allows a formulation in terms of the Coxeter group $A_3$. This short note includes miscellaneous remarks on the generalizations along $B_3, C_3, F_4$ and…
In \cite{Sz17-2} we considered hyperball packings in $3$-dimensional hyperbolic space. We developed a decomposition algorithm that for each saturated hyperball packing provides a decomposition of $\HYP$ into truncated tetrahedra. In order…
The main result of this paper describes the normalizer of a finite parabolic subgroup of a (possibly infinite) Coxeter group. We use this to compute the automorphism groups of some Lorentzian lattices and K3 surfaces.