Hyperbolization of polyhedra

Relative Hyperbolization and Aspherical Bordisms: an Addendum to “hyperbolization of Polyhedra”

We give two versions of relative hyperbolization. We use the first version to prove that if (each component of) a closed manifold M is aspherical and if M is a boundary, then it is the boundary of an aspherical manifold.

Hyperbolization of Euclidean Ornaments

In this article we outline a method that automatically transforms an Euclidean ornament into a hyperbolic one. The necessary steps are pattern recognition, symmetry detection, extraction of a Euclidean fundamental region, conformal deformation to a hyperbolic fundamental region and tessellation of the hyperbolic plane with this patch. Each of these steps has its own mathematical subtleties that...

Graphs of polyhedra; polyhedra as graphs

Relations between graph theory and polyhedra are presented in two contexts. In the first, the symbiotic dependence between 3-connected planar graphs and convex polyhedra is described in detail. In the second, a theory of nonconvex polyhedra is based on a graph-theoretic foundation. This approach eliminates the vagueness and inconsistency that pervade much of the literature dealing with polyhedr...

Weak Hyperbolization Conjecture for 3-dimensional Cat(0) Groups

In this note we prove a weak hyperbolization conjecture for CAT(0) 3-dimensional Poincaré duality groups.

polyhedra ∗