Compact hyperbolic Coxeter four-polytopes with eight facets

Deformation spaces of Coxeter truncation polytopes

A convex polytope $P$ in the real projective space with reflections in the facets of $P$ is a Coxeter polytope if the reflections generate a subgroup $Γ$ of the group of projective transformations so that the $Γ$-translates of the interior of $P$ are mutually disjoint. It follows from work of Vinberg that if $P$ is a Coxeter polytope, then the interior $Ω$ of the $Γ$-orbit of $P$ is convex and $Γ$ acts properly discontinuously on $Ω$. A Coxeter polytope $P$ is $2$-perfect if $P \smallsetminus Ω$ consists of only some vertices of $P$. In this paper, we describe the deformation spaces of $2$-per…

二完美截断多面体上的整数表示的有限性

Finiteness of integral representations on 2-perfect truncation polytopes

研究二完美截断多面体的整数表示有限性，证明其变形空间仅含有限多个整数表示，扩展到一般不可约、大、二完美截断多面体。

Lannér diagrams and combinatorial properties of compact hyperbolic Coxeter polytopes

In this paper we study $\times_0$-products of Lannér diagrams. We prove that every $\times_0$-product of at least four Lannér diagrams with at least one diagram of order $\ge 3$ is superhyperbolic. As a corollary, we obtain that known classifications exhaust all compact hyperbolic Coxeter polytopes that are combinatorially equivalent to products of simplices. We also consider compact hyperbolic Coxeter polytopes whose every Lannér subdiagram has order $2$. The second result of this paper slightly improves recent Burcroff's upper bound on the dimension of such polytopes to $12$.

On faces of quasi-arithmetic Coxeter polytopes

We prove that each lower-dimensional face of a quasi-arithmetic Coxeter polytope, which happens to be itself a Coxeter polytope, is also quasi-arithmetic. We also provide a sufficient condition for a codimension $1$ face to be actually arithmetic, as well as a few computed examples.

On volumes of hyperbolic Coxeter polytopes and quadratic forms

In this paper, we compute the covolume of the group of units of the quadratic form f_d^n(x) = x_1^2 + x_2^2 + . . . + x_n^2 - d x_{n+1}^2 with d an odd, positive, square-free integer. Mcleod has determined the hyperbolic Coxeter fundamental domain of the reflection subgroup of the group of units of the quadratic form f_3^n. We apply our covolume formula to compute the volumes of these hyperbolic Coxeter polytopes.

From geometry to arithmetic of compact hyperbolic Coxeter polytopes

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 arithmetic hyperbolic reflection groups.

几类不可公度的非紧双曲考克斯特多面体

Several families of incommensurable noncompact hyperbolic Coxeter polytopes

研究对141个有八个面的有限体积双曲考克斯特五维多面体分类，利用特定方法在4至9维构造无限多个两两不可公度的非紧考克斯特多面体，可公度类数量至少随体积指数增长。

Locally rigid right-angled Coxeter groups with Fuchsian ends in dimension 5

In this paper, we construct a right-angled 5-polytope P of finite volume such that all the right-angled Coxeter groups with Fuchsian ends obtained from P are locally rigid.

Near Classification of Compact Hyperbolic Coxeter $d$-Polytopes with $d+4$ Facets and Related Dimension Bounds

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 method for generating the combinatorial type of these polytopes via the classification of point set order types. In dimensions $4$ and $5$, there are $348$ and $51$ polytopes, respectively, yielding many new examples for further study. We furthermore provide new upper bounds on the dimension $d$ of compact hyperbolic Coxeter polytopes with $d+k$ facets for $…

Projective deformations of weakly orderable hyperbolic Coxeter orbifolds

A Coxeter $n$-orbifold is an $n$-dimensional orbifold based on a polytope with silvered boundary facets. Each pair of adjacent facets meet on a ridge of some order $m$, whose neighborhood is locally modeled on ${\mathbb R}^n$ modulo the dihedral group of order $2m$ generated by two reflections. For $n \geq 3$, we study the deformation space of real projective structures on a compact Coxeter $n$-orbifold $Q$ admitting a hyperbolic structure. Let $e_+(Q)$ be the number of ridges of order $\geq 3$. A neighborhood of the hyperbolic structure in the deformation space is a cell of dimension $e_+(Q) …

A convexity theorem for real projective structures

Given a finite collection P of convex n-polytopes in RP^n (n>1), we consider a real projective manifold M which is obtained by gluing together the polytopes in P along their facets in such a way that the union of any two adjacent polytopes sharing a common facet is convex. We prove that the real projective structure on M is (1) convex if P contains no triangular polytope, and (2) properly convex if, in addition, P contains a polytope whose dual polytope is thick. Triangular polytopes and polytopes with thick duals are defined as analogues of triangles and polygons with at least five edges, res…

有限体积的射影反射群

Projective reflection groups of finite covolume

该研究证明了有限体积的Coxeter多面体与Vinberg域的关系，指出其在反射群中的不变性条件。

Perfect but not generating Delaunay polytopes

In his seminal 1951 paper "Extreme forms" Coxeter \cite{cox51} observed that for $n \ge 9$ one can add vectors to the perfect lattice $\sfA_9$ so that the resulting perfect lattice, called $\sfA_9^2$ by Coxeter, has exactly the same set of minimal vectors. An inhomogeneous analog of the notion of perfect lattice is that of a lattice with a perfect Delaunay polytope: the vertices of a perfect Delaunay polytope are the analogs of minimal vectors in a perfect lattice. We find a new infinite series $P(n,s)$ for $s\geq 2$ and $n+1\geq 4s$ of $n$-dimensional perfect Delaunay polytopes. A remarkable …

Buchstaber Invariant of Simple Polytopes

In this paper we study a new combinatorial invariant of simple polytopes, which comes from toric topology. With each simple n-polytope P with m facets we can associate a moment-angle complex Z_P with a canonical action of the torus T^m. Then s(P) is the maximal dimension of a toric subgroup that acts freely on Z_P. The problem stated by Victor M. Buchstaber is to find a simple combinatorial description of an s-number. We describe the main properties of s(P) and study the properties of simple n-polytopes with n+3 facets. In particular, we find the value of an s-number for such polytopes, a simp…

论双曲空间中的ADEG多面体

On ADEG-polyhedra in hyperbolic spaces

本文证明高维双曲Coxeter多面体的非零二面角存在下界，并通过对具有指定二面角的多面体进行构造性分类，完整刻画了仅含二面角π/2、π/3、π/6且无不相交面的ADEG多面体，发现一个新的9维14面多面体。