Deformation spaces of Coxeter truncation polytopes

Abstract

A convex polytope P $P$ in the real projective space with reflections in the facets of P $P$ is a Coxeter polytope if the reflections generate a subgroup Γ $\Gamma$ of the group of projective transformations so that the Γ $\Gamma$ -translates of the interior of P $P$ are mutually disjoint. It follows from work of Vinberg that if P $P$ is a Coxeter polytope, then the interior Ω $\Omega$ of the Γ $\Gamma$ -orbit of P $P$ is convex and Γ $\Gamma$ acts properly discontinuously on Ω $\Omega$ . A Coxeter polytope P $P$ is 2 $\hskip.001pt 2$ -perfect if P ∖ Ω $P \smallsetminus \Omega$ consists of only some vertices of P $P$ . In this paper, we describe the deformation spaces of 2 $\hskip.001pt 2$ -perfect Coxeter polytopes P $P$ of dimensions d ⩾ 4 $d \geqslant 4$ with the same dihedral angles when the underlying polytope of P $P$ is a truncation polytope, that is, a polytope obtained from a simplex by successively truncating vertices. The deformation spaces of Coxeter truncation polytopes of dimensions d = 2 $d = 2$ and d = 3 $d = 3$ were studied, respectively, by Goldman and the third author.