Constructing subdivision rules from polyhedra with identifications

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 finite subdivision rule for many $3$–manifolds obtained from polyhedral gluings. The manifolds that satisfy the conditions include all manifolds created from compact right angled hyperbolic polyhedra, as well as many $3$–manifolds with toral or hyperbolic boundary.