Related papers: Coxeter polytopes with a unique pair of non-inters…

In this thesis, I am going to consider Bernoulli percolation on graphs admitting vertex-transitive actions of groups of isometries of d-dimensional hyperbolic spaces H^d. In the first chapter, I give an overview concerning percolation and…

For any Coxeter system $(W,S)$ of rank $n$, we introduce an abstract boolean complex (simplicial poset) of dimension $2n-1$ that contains the Coxeter complex as a relative subcomplex. Faces are indexed by triples $(I,w,J)$, where $I$ and…

One can define what it means for a compact manifold with corners to be a "contractible manifold with contractible faces." Two combinatorially equivalent, contractible manifolds with contractible faces are diffeomorphic if and only if their…

Wythoff's construction associates a uniform polytope to a Coxeter diagram whose vertices are decorated with crosses, which indicate the subgroup stabilizing a generic point. Champagne, Kjiri, Patera, and Sharp remarked that by associating…

Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its $1$-skeleton. Call a vertex of a $d$-polytope \emph{nonsimple} if the number of edges incident to it is more than $d$.…

The Wythoff construction takes a $d$-dimensional polytope $P$, a subset $S$ of $\{0,..., d\}$ and returns another $d$-dimensional polytope $P(S)$. If $P$ is a regular polytope, then $P(S)$ is vertex-transitive. This construction builds a…

We describe a provably complete algorithm for the generation of a tight, possibly exact superset of all combinatorially distinct simple n-facet polytopes in R^d, along with their graphs, f-vectors, and face lattices. The technique applies…

Let P be a d-dimensional lattice polytope. We show that there exists a natural number c_d, only depending on d, such that the multiples cP have a unimodular cover for every natural number c >= c_d. Actually, a subexponential upper bound for…

Any convex polytope whose combinatorial automorphism group has two orbits on the flags is isomorphic to one whose group of Euclidean symmetries has two orbits on the flags (equivalently, to one whose automorphism group and symmetry group…

A closed connected hyperbolic $n$-manifold bounds geometrically if it is isometric to the geodesic boundary of a compact hyperbolic $(n+1)$-manifold. A. Reid and D. Long have shown by arithmetic methods the existence of infinitely many…

We prove a lower bound theorem for the number of $k$-faces ($1\le k\le d-2$) in a $d$-dimensional polytope $P$ (or $d$-polytope) with up to $3d-1$ vertices. Previous lower bound theorems for $d$-polytopes with few vertices concern those…

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…

The smallest three hyperbolic compact arithmetic 5-orbifolds can be derived from two compact Coxeter polytops which are combinatorially simplicial prisms (or complete orthoschemes of degree $d=1$) in the five dimensional hyperbolic space…

A polygonal surface in the pseudo-hyperbolic space H^(2,n) is a complete maximal surface bounded by a lightlike polygon in the Einstein universe Ein^(1,n) with finitely many vertices. In this article, we give several characterizations of…

An abstract polytope is \emph{flat} if every facet is incident on every vertex. In this paper, we prove that no chiral polytope has flat finite regular facets and finite regular vertex-figures. We then determine the three smallest non-flat…

K3 polytopes appear in complements of tropical quartic surfaces. They are dual to regular unimodular central triangulations of reflexive polytopes in the fourth dilation of the standard tetrahedron. Exploring these combinatorial objects, we…

Unlike the situation in the classical theory of convex polytopes, there is a wealth of semi-regular abstract polytopes, including interesting examples exhibiting some unexpected phenomena. We prove that even an equifacetted semi-regular…

A Coxeter polytope is a convex polytope in a real projective space equipped with linear reflections in its facets, such that the orbits of the polytope under the action of the group generated by the linear reflections tessellate a convex…

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…

We conjecture that for every dimension n not equal 3 there exists a noncompact hyperbolic n-manifold whose volume is smaller than the volume of any compact hyperbolic n-manifold. For dimensions n at most 4 and n=6 this conjecture follows…