Projective deformations of weakly orderable hyperbolic Coxeter orbifolds

Abstract

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 ${ℝ}^n$ modulo the dihedral group of order $\mathfrak{2}m$ generated by two reflections. For $n\ge \mathfrak{3}$, we study the deformation space of real projective structures on a compact Coxeter $n$–orbifold $Q$ admitting a hyperbolic structure. Let $e_{+}\left(Q\right)$ be the number of ridges of order greater than or equal to $\mathfrak{3}$. A neighborhood of the hyperbolic structure in the deformation space is a cell of dimension $e_{+}\left(Q\right)-n$ if $n=\mathfrak{3}$ and $Q$ is weakly orderable, ie the faces of $Q$ can be ordered so that each face contains at most $\mathfrak{3}$ edges of order $\mathfrak{2}$ in faces of higher indices, or $Q$ is based on a truncation polytope.