From the hyperbolic 24–cell to the cuboctahedron

Abstract

Abstract. We describe a family of 4-dimensional hyperbolic orbifolds, constructed by deforming an infinite volume orbifold obtained from the ideal, hyperbolic 24-cell by removing two walls. This family provides an infinite number of infinitesimally rigid, infinite covolume, geometrically finite discrete subgroups of Isom(H 4 ). It also leads to finite covolume Coxeter groups which are the homomorphic image of the group of reflections in the hyperbolic 24-cell. The examples are constructed very explicitly, both from an algebraic and a geometric point of view. The method used can be viewed as a 4-dimensional, but infinite volume, analog of 3-dimensional hyperbolic Dehn filling.

New hyperbolic 4–manifolds of low volume

“…In the case of the negative facets, the pairing map is given by the involution i defined by (7). This induces an orientation-preserving involution of the figure-eight knot complement which is, however, not fixed-point-free.…”

Infinitesimal rigidity of a compact hyperbolic 4–orbifold with totally geodesic boundary

“…Note that Conjecture 1.1 is false without the assumption that is convex cocompact. A counterexample was studied in [5]. As a first step toward proving this conjecture, our goal is to verify it in a specific 4-dimensional example.…”

“…Even weakening the convex cocompact condition to geometric finiteness makes the conjecture false. A counterexample is presented in [5]. Nonetheless, it seems reasonable to expect that Conjecture 1.1 is true.…”

“…As a first step in this direction, Kerckhoff and Storm studied deformations of a specific discrete subgroup of Isom.H 4 /, the isometry group of hyperbolic 4-space [5]. Of particular interest in this study were groups with Fuchsian ends that are not Fuchsian, meaning groups < Isom.H n / whose convex hull is n-dimensional, has finite volume and has totally geodesic boundary.…”

Hyperbolic Dehn filling in dimension four

“…The construction goes as follows. The fundamental ingredient is a deforming family F t ⊂ H 4 of infinite-volume polytopes built by Kerckhoff and Storm in [14]. We truncate here F t via two additional hyperplanes to get a deforming family of finite-volume polytopes P t ⊂ H 4 .…”