Projective deformations of hyperbolic Coxeter 3-orbifolds

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.

The convex real projective orbifolds with radial or totally geodesic ends: A survey of some partial results

“…The interior of a finite-volume hyperbolic n-orbifold with rank (n − 1) horospherical ends and totally geodesic boundary forms an example of a properly convex strongly tame real projective orbifold with radial or totally geodesic ends. and their deformations are computed in [28]. Also, see Ryan Greene [41] for the theory.…”

“…(See Chapter 7 of [19].) Except for ones based on tetrahedra, complete hyperbolic Coxeter 3-orbifolds with all edge orders 3 have at least six dimensional deformation spaces by Theorem 1 of Choi-Hodgson-Lee [28].…”

“…Benoist [10] first found such phenomena for a Coxeter 3-orbifold. We also had the many numerical and theoretical results for above Coxeter 3-orbifolds which we plan to write more explicitly in a later paper [27]. Also, Greene [41] found many such examples using explicit computations.…”

“…More specifically, we can consider a hyperbolic ideal simplex or a hyperbolic ideal cube with such structures. (See Choi-Hodgson-Lee [28] for examples of 6-dimensional deformations. )…”

“…Originally, we developed the theory for orbifolds as given in papers of Choi [24], Choi, Hodgson, and Lee [28], and [19]. However, the recent examples of Ballas [1], [2], and Ballas, Danciger, and Lee [4] can be covered using fixing sections.…”