Creating subdivision rules from polyhedra with identifications

TL;DR

This paper constructs explicit subdivision rules for a broad class of 3-manifolds derived from polyhedral gluings, advancing the understanding of subdivision rules related to hyperbolic groups.

Contribution

It provides the first explicit subdivision rules for many 3-manifolds from polyhedral gluings, including those from right-angled hyperbolic polyhedra.

Findings

Explicit subdivision rules for numerous 3-manifolds

Applicable to manifolds with hyperbolic or toral boundaries

Enhances understanding of subdivision rules in hyperbolic geometry

Abstract

Cannon, Swenson, and others have proved numerous theorems about subdivision rules associated to hyperbolic groups with a 2-sphere at infinity. However, few explicit examples are known. We construct an explicit subdivision rule for many 3-manifolds from polyhedral gluings. The manifolds that satisfy the conditions include all closed manifolds created from right-angled hyperbolic polyhedra, as well as many 3-manifolds with toral or hyperbolic boundary.