Deformation spaces of Coxeter truncation polytopes

TL;DR

This paper investigates the deformation spaces of 2-perfect Coxeter polytopes, specifically truncation polytopes, in dimensions four and higher, extending previous work on lower-dimensional cases.

Contribution

It characterizes the deformation spaces of high-dimensional Coxeter truncation polytopes with fixed dihedral angles, generalizing earlier results for lower dimensions.

Findings

Deformation spaces are described for 2-perfect Coxeter truncation polytopes in dimensions ≥ 4.

The work extends the understanding of Coxeter polytopes beyond dimensions 2 and 3.

New classifications of deformation spaces are provided for these polytopes.

Abstract

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 $\Gamma$ of the group of projective transformations so that the $\Gamma$-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 $\Omega$ of the $\Gamma$-orbit of $P$ is convex and $\Gamma$ acts properly discontinuously on $\Omega$. A Coxeter polytope $P$ is $2$-perfect if $P \smallsetminus \Omega$ consists of only some vertices of $P$. In this paper, we describe the deformation spaces of $2$-perfect Coxeter polytopes $P$ of dimension $d \geqslant 4$ with the same dihedral angles when the underlying polytope of $P$ is a truncation polytope, i.e. a polytope obtained from a simplex by successively truncating vertices. The deformation spaces of Coxeter truncation polytopes of dimension $d = 2$ and $d = 3$ were studied respectively by Goldman and the third author.