Hyperbolic four-manifolds

Abstract

This is a short survey on finite-volume hyperbolic four-manifolds. We first describe some general theorems, and then focus on the geometry of the concrete examples that we found in the literature.The starting point of most constructions is an explicit reflection group Γ acting on H 4 , together with its Coxeter polytope P . Hyperbolic manifolds then arise either algebraically from the determination of torsion-free subgroups of Γ, or more geometrically by assembling copies of P .We end the survey by raising a few open questions.

Golden codes: quantum LDPC codes built from regular tessellations of hyperbolic 4-manifolds

“…It is possible to generalise the definition of the Euler characteristic (see e.g. [26]) to orbifolds (roughly speaking, manifolds that can have singularities) in such a way that this definition still holds. We can illustrate this by computing the Euler characteristic of the hypercube T of the {4,3,3,5} tessellation of hyperbolic 4-space.…”