Related papers: Essential hyperbolic Coxeter polytopes

We classify the polycyclic totally ordered simple dimension groups, i.e. dimension groups given by a dense embedding of n-dimensional lattice into the real line. Our method is based on the geometry of simple geodesics on the hyperbolic…

We establish necessary and sufficient conditions for the realization of mapping schemata as post-critically finite polynomials, or more generally, as post-critically finite polynomial maps from a finite union of copies of the complex…

A (convex) polytope $P$ is said to be $2$-level if for every direction of hyperplanes which is facet-defining for $P$, the vertices of $P$ can be covered with two hyperplanes of that direction. The study of these polytopes is motivated by…

In this paper we consider convex co-compact subgroups of the projective linear group. We prove that such a group is relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank two if and only if each open face…

Let $(W,S)$ be a Coxeter system with $I\subseteq S$ such that the parabolic subgroup $W_I$ is finite. Associated to this data there is a \textit{Hecke algebra} $\scH$ and a \textit{parabolic Hecke algebra}…

We give formulas for the second and third integral homology of an arbitrary finitely generated Coxeter group, solely in terms of the corresponding Coxeter diagram. The first of these calculations refines a theorem of Howlett, while the…

This paper considers the planar figure of a combinatorial polytope or tessellation identified by the Coxeter symbol $k_{i,j}$ , inscribed in a conic, satisfying the geometric constraint that each octahedral cell has a centre. This…

We classify here combinatorially rigid simple polytopes with three facets more than their dimension.

In $n$-dimensional hyperbolic space $\mathbf{H}^n$ $(n\ge2)$ there are $3$-types of spheres (balls): the sphere, horosphere and hypersphere. If $n=2,3$ we know an universal upper bound of the ball packing densities, where each ball volume…

Kepler (1619) and Croft (1980) have considered largest homothetic copies of one regular polytope contained in another regular polytope. For arbitrary pairs of polytopes we propose to model this as a quadratically constrained optimization…

Sometimes, it is possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope $P$ is defined to be the minimum number of facets of a (possibly…

Let ${\mathcal A}$ be a finite real linear hyperplane arrangement in three dimensions. Suppose further that all the regions of ${\mathcal A}$ are isometric. We prove that ${\mathcal A}$ is necessarily a Coxeter arrangement. As it is well…

We prove that each special Lorentzian holonomy group (with the exception of those including the isotropy groups of K\"ahler symmetric spaces with rank greater than one) can be realized as the holonomy group of a globally hyperbolic…

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…

A compact hyperbolic "cobweb" manifold (hyperbolic space form) of symbol $Cw(6,6,6)$ will be constructed in Fig.1,4,5 as a representant of a presumably infinite series $Cw(2p,2p,2p)$ $(3 \le p \in \bN$ natural numbers). This is a by-product…

Here we prove classification results announced in Part I (alg-geom/9711032). We classify maximal hyperbolic root systems of the rank 3 having restricted arithmetic type and a generalized lattice Weyl vector $\rho$ with $\rho^2\ge 0$ (i.e.…

We introduce and study a combinatorially defined notion of root basis of a (real) root system of a possibly infinite Coxeter group. Known results on conjugacy up to sign of root bases of certain irreducible finite rank real root systems are…

A finite subgroup of $GL(n,\mathbb C)$ is involutory if the sum of the dimensions of its irreducible complex representations is given by the number of absolute involutions in the group. A uniform combinatorial model is constructed for all…

Consider polynomial maps $f:\C\to\C$ of degree $d\ge 2$, or more generally polynomial maps from a finite union of copies of $\C$ to itself. In the space of suitably normalized maps of this type, the hyperbolic maps form an open set called…

Simplicial arrangements are classical objects in discrete geometry. Their classification remains an open problem but there is a list conjectured to be complete at least for rank three. A further important class in the theory of hyperplane…