Veech Groups and Triangulations of Half-Dilation Pillowcases

TL;DR

This paper explores the symmetries of half-dilation structures on spheres with four singularities, demonstrating a tetrahedral construction for all such surfaces and computing their Veech groups.

Contribution

It introduces a tetrahedral construction method for all triangulable half-dilation pillowcases and calculates their Veech groups, advancing understanding of their symmetry properties.

Findings

All such surfaces can be generated by a tetrahedral construction.

The symmetry groups (Veech groups) are explicitly computed.

The structure of these groups provides insight into the geometric symmetries of the surfaces.

Abstract

In this paper we consider the symmetries of triangulable half-dilation structures on the sphere with four singularities. We show that all such surfaces can be produced by a tetrahedral construction. Using this construction, we calculate each such surface's symmetry group in $\operatorname{PSL}(2, \mathbb{R})$ called the Veech group.