Character varieties of a transitioning Coxeter 4-orbifold

TL;DR

This paper investigates the character varieties of a specific 4-orbifold associated with a Coxeter group, exploring geometric transitions between hyperbolic, Anti-de Sitter, and half-pipe geometries and analyzing singularities at collapse points.

Contribution

It extends the understanding of geometric transitions in 4-orbifolds by analyzing character varieties across different geometries and studying rigidity properties of cusp groups in four dimensions.

Findings

Character varieties exhibit specific singularities at collapse points.

Transitions between hyperbolic, Anti-de Sitter, and half-pipe geometries are characterized.

Rigidity properties of 4-dimensional cusp groups influence the structure of character varieties.

Abstract

In 2010, Kerckhoff and Storm discovered a path of hyperbolic 4-polytopes eventually collapsing to an ideal right-angled cuboctahedron. This is expressed by a deformation of the inclusion of a discrete reflection group (a right-angled Coxeter group) in the isometry group of hyperbolic 4-space. More recently, we have shown that the path of polytopes can be extended to Anti-de Sitter geometry so as to have geometric transition on a naturally associated 4-orbifold, via a transitional half-pipe structure. In this paper, we study the hyperbolic, Anti-de Sitter, and half-pipe character varieties of Kerckhoff and Storm's right-angled Coxeter group near each of the found holonomy representations, including a description of the singularity that appears at the collapse. An essential tool is the study of some rigidity properties of right-angled cusp groups in dimension four.