Isotopic tiling theory for hyperbolic surfaces

TL;DR

This paper develops mathematical tools to classify and enumerate isotopy classes of tilings on hyperbolic surfaces, extending combinatorial tiling theory through orbifold mapping class groups.

Contribution

It generalizes Delaney-Dress tiling theory to hyperbolic surfaces with boundary and punctures using orbifold mapping class groups, enabling complete enumeration of tilings.

Findings

Extended tiling classification to nonorientable surfaces

Developed finite symbol encoding for tiling complexity

Provided framework for enumerating isotopically distinct tilings

Abstract

In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.