Projective deformations of weakly orderable hyperbolic Coxeter orbifolds

TL;DR

This paper investigates the deformation space of real projective structures on hyperbolic Coxeter orbifolds, revealing conditions under which this space forms a cell of specific dimension related to the orbifold's ridges and order properties.

Contribution

It characterizes the local structure of the deformation space for hyperbolic Coxeter orbifolds in terms of weak orderability and polytope truncation, extending understanding of their geometric deformations.

Findings

Neighborhood of hyperbolic structure is a cell of dimension e_+(Q) - n for n=3 under certain conditions.

Deformation space dimension depends on the number of ridges with order ≥ 3.

Weak orderability or truncation polytope basis ensures a well-behaved deformation space.

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 ${\mathbb R}^n$ 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 $e_+(Q) - n$ if $n=3$ and $Q$ is weakly orderable, i.e., the faces of $Q$ can be ordered so that each face contains at most $3$ edges of order $2$ in faces of higher indices, or $Q$ is based on a truncation polytope.