Grid graphs and lattice surfaces

TL;DR

This paper explores the construction and properties of translation surfaces derived from grid graphs using Thurston's method, analyzing their Veech groups, and identifying limitations on which triangle groups can occur as Veech groups.

Contribution

It introduces a new approach to constructing these surfaces, computes their Veech groups, and extends understanding of which triangle groups can serve as Veech groups.

Findings

Constructed translation surfaces from grid graphs with specific Veech groups

Computed Veech groups and addressed primitivity questions

Identified certain triangle groups that cannot be Veech groups

Abstract

First, we apply Thurston's construction of pseudo-Anosov homeomorphisms to grid graphs and obtain translation surfaces whose Veech groups are commensurable to $(m,n,\infty)$ triangle groups. These surfaces were first discovered by Bouw and M\"oller, however our treatment of the surfaces differs. We construct these surfaces by gluing together polygons in two ways. We use these elementary descriptions to compute the Veech groups, resolve primitivity questions, and describe the surfaces algebraically. Second, we show that some $(m,n, \infty)$ triangle groups can not arise as Veech groups. This generalizes work of Hubert and Schmidt.