Veech Groups and Triangulations of Half-Dilation Pillowcases

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 $\mathrm{PSL}(2,\mathbb{R})$ called the Veech group.

17 pages, 10 figures