From Convex Ideal Polyhedra to Fundamental Domains in H^3

TL;DR

This paper explores the relationship between polyhedra and groups in hyperbolic 3-space, providing formulas, classifications, and restrictions that deepen understanding of fundamental domains and their associated groups.

Contribution

It introduces a formula for edge classes in torsion-free groups and classifies fundamental domains on the cube, linking polyhedral properties with group characteristics.

Findings

Derived a formula for edge classes in torsion-free groups

Classified all fundamental domains on the cube with torsion-free groups

Established restrictions on edge classes based on group properties

Abstract

Our goal is to better understand the relationship between the polyhedron and the group associated with a fundamental domain in H^3. In this paper, we will study torsion-free groups and determine a formula for how many edge classes a given abstract polyhedron must have. We will use that result to classify all fundamental domains on the cube with torsion-free groups, including a discussion of the explicit groups associated to those domains. We will then turn to more general fundamental domains and prove a series of results about how properties of the group place restrictions on the edge classes in the quotient manifold. These results give insight into how the polyhedron and the group associated to a fundamental domain interact, as well as offering concrete tools to find fundamental domains.