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 Gamma$\Gamma$ of the group of projective transformations so that the Gamma$\Gamma$-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 omega$\Omega$ of the Gamma$\Gamma$-orbit of P$P$ is convex and Gamma$\Gamma$ acts properly discontinuously on omega$\Omega$. A Coxeter polytope P$P$ is 2$\hskip.001pt 2$-perfect if P set minus omega$P \smallsetminus \Omega$ 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.