On the Characterization of Polyhedra in Hyperbolic 3-Space

TL;DR

This paper reviews key results on the characterization of polyhedra in hyperbolic 3-space, including Rivin's and Hodgson's theorems, and discusses limitations in determining polyhedra solely by edge lengths.

Contribution

It synthesizes existing theorems on hyperbolic polyhedra and highlights the limitations of edge length determination through counter-examples.

Findings

Rivin's theorem characterizes compact convex hyperbolic polyhedra.

Hodgson proved Adreev's theorem in hyperbolic space.

Counter-examples show hyperbolic polyhedra are not uniquely determined by edge lengths.

Abstract

We review several results related to the characterization of polyhedra in hyperbolic 3-space. In particular we present Rivin's theorem that gives a characterization of compact convex hyperbolic polyhedra, and Hodgson's proof of the Adreev's theorem. We also review the analogous characterization of ideal polyhedra, and give a family of counter-examples that proves that hyperbolic polyhedra are not determined by edge lengths.