Projective Deformations of Hyperbolic Coxeter 3-Orbifolds

TL;DR

This paper explores how certain hyperbolic structures on 3-orbifolds can deform into more general real projective structures, revealing new classes of deformations and providing numerical and exact results for specific polyhedral cases.

Contribution

It identifies new classes of hyperbolic reflection 3-orbifolds that admit projective deformations beyond hyperbolic structures, expanding understanding of geometric structures on orbifolds.

Findings

Existence of deformations for certain hyperbolic 3-orbifolds.

Numerical results on projective deformations of hyperbolic cubes.

Exact results on deformations of hyperbolic dodecahedra.

Abstract

By using Klein's model for hyperbolic geometry, hyperbolic structures on orbifolds or manifolds provide examples of real projective structures. By Andreev's theorem, many 3-dimensional reflection orbifolds admit a finite volume hyperbolic structure, and such a hyperbolic structure is unique. However, the induced real projective structure on some such 3-orbifolds deforms into a family of real projective structures that are not induced from hyperbolic structures. In this paper, we find new classes of compact and complete hyperbolic reflection 3-orbifolds with such deformations. We also explain numerical and exact results on projective deformations of some compact hyperbolic cubes and dodecahedra.