Related papers: Injectivity radii of hyperbolic polyhedra

We compute Coxeter diagrams of several ``large'' reflective even 2-elementary hyperbolic lattices and their maximal parabolic subdiagrams, and give some applications of these results to the theory of K3 surfaces and hyperkahler varieties.

For a closed, strictly convex projective manifold of dimension $n\geq 3$ that admits a hyperbolic structure, we show that the ratio of Hilbert volume to hyperbolic volume is bounded below by a constant that depends only on dimension. We…

We prove for the first time a pointwise lower estimate of the normal injectivity radius of an embedded hypersurface in an arbitrary Riemannian manifold. Main applications include: (i) a pointwise lower estimate of the graphing radius of a…

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…

We study rings which have Noetherian cohomology under the action of a ring of cohomology operators. The main result is a criterion for a complex of modules over such a ring to have finite injective dimension. This criterion generalizes, by…

We observe that fullerene graphs are one-skeletons of polyhedra, which can be realized with all dihedral angles equal to $\pi/2$ in a hyperbolic 3-dimensional space. One of the most important invariants of such a polyhedron is its volume.…

The Hessian of the renormalized volume of geometrically finite hyperbolic $3$-manifolds without rank-$1$ cusps, computed at the hyperbolic metric $g$ with totally geodesic boundary of the convex core, is shown to be a strictly positive…

Following Vinberg, we find the criterions for a subgroup generated by reflections $\Gamma \subset \SL^{\pm}(n+1,\mathbb{R})$ and its finite-index subgroups to be definable over $\mathbb{A}$ where $\mathbb{A}$ is an integrally closed…

We consider triangle faced convex polyhedra inscribed in the unit sphere $S^2$ in ${\Bbb{R}}^3$. One way of measuring their deviation from regular polyhedra with triangular faces is to consider the quotient of the lengths of the longest and…

In this paper, we obtain a complete classification of compact hyperbolic Coxeter five-dimensional polytopes with nine facets.

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…

We show that the infimum of the dual volume of the convex core of a convex co-compact hyperbolic $3$-manifold with incompressible boundary coincides with the infimum of the Riemannian volume of its convex core, as we vary the geometry by…

We use methods of combinatorics of polytopes together with geometrical and computational ones to obtain the complete list of compact hyperbolic Coxeter n-polytopes with n+3 facets, 3<n<8. Combined with results of Esselmann (1994), Andreev…

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…

This paper investigates the relationship between the topology of hyperbolizable 3-manifolds M with incompressible boundary and the volume of hyperbolic convex cores homotopy equivalent to M. Specifically, it proves a conjecture of Bonahon…

We determine the optimal horoball packing densities for Koszul-type Coxeter simplex tilings in hyperbolic $3$-space. Using a parametrization of horoballs by the Busemann function and the symmetry of the tilings, we obtain families of…

In this paper we derive an explicit lower bound on the volume of a hyperbolic $n$-orbifold for dimensions greater than or equal to four. Our main tool is H. C. Wang's bound on the radius of a ball embedded in the fundamental domain of a…

We prove that, apart from some well-known low-dimensional examples, any compact hyperbolic Coxeter polytope has a pair of disjoint facets. This is one of very few known general results concerning combinatorics of compact hyperbolic Coxeter…

In this paper, we present the first nontrivial lower bound on the translative covering density of tetrahedra. To this end, we show the lower bound, in any translative covering of tetrahedra, on the density relative to a given cube. The…

In this paper we determine new upper bounds for the maximal density of translative packings of superballs in three dimensions (unit balls for the $l^p_3$-norm) and of Platonic and Archimedean solids having tetrahedral symmetry. Thereby, we…