Subdivision rules and the eight model geometries

TL;DR

This paper extends the concept of subdivision rules to various 3-manifold geometries, providing explicit rules for several types and visualizing their actions at infinity, thereby broadening understanding of geometric group actions.

Contribution

It introduces explicit finite subdivision rules for multiple 3-manifold geometries, including E3, H2XR, S2XR, S3, and SL2(R), expanding previous results beyond hyperbolic cases.

Findings

Subdivision rules for several geometries are explicitly constructed.

Subdivision rules for these geometries are similar across finite covers.

Visualization of subdivision rules at infinity using Circlepack.

Abstract

Cannon and Swenson have shown that each hyperbolic 3-manifold group has a natural subdivision rule on the space at infinity, and that this subdivision rule captures the action of the group on the sphere. Explicit subdivision rules have also been found for some closed and finite-volume hyperbolic manifolds, as well as a few non-hyperbolic knot complements. We extend these results by finding explicit finite subdivision rules for closed manifolds of the E3, H2XR, S2XR, S3, and SL2(R) manifolds by examining model manifolds. Because all manifolds in these geometries are the same up to finite covers, the subdivision rules for these model manifolds will be very similar to subdivision rules for all other manifolds in their respective geometries. We also discuss the existence of subdivision rules for Nil and Sol geometries. We use Ken Stephenson's Circlepack to visualize the subdivision rules and the resulting space at infinity.