Grid graphs and lattice surfaces

By Patrick Hooper. 2008

51 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8da950c96f0fa158796b8ec8a913aab36763b22d

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In Computational Theory and Mathematics

1. Kasra Rafi
2. Kathryn Mann

In Geometry and Topology

1. Kasra Rafi
2. Kathryn Mann
3. Samuel J. Taylor
4. U. Hamenstaedt
5. Saul Schleimer

In Mathematical Physics

1. Kasra Rafi
2. Kathryn Mann
3. U. Hamenstaedt

Citations in the corpus

• Large-scale geometry of big mapping class groups (2019, background): “To our knowledge, tame surfaces include all concrete examples studied thus far in the literature, including the mapping class groups of some specific infinite type surfaces in [3, 2, 8], and the discussion of geometric or dynamical properties of various translation surfaces of infinite type in [6, 9”