Grid Graphs and Lattice Surfaces

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,∞) triangle groups. Many (if not all) of the surfaces we construct were first discovered by Bouw and Möller, 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,∞) triangle groups can not arise as Veech groups. This generalizes work of Hubert and Schmidt.