Constructing lattice surfaces with prescribed Veech groups: an algorithm

Research output: Working paperPreprintAcademic

Abstract

The Veech group of a translation surface is the group of Jacobians of orientation-preserving affine automorphisms of the surface. We present an algorithm which constructs all translation surfaces with a given lattice Veech group in any given stratum. In developing this algorithm, we give a new proof of a finiteness result of Smillie and Weiss, namely that there are only finitely many (unit-area) translation surfaces in any stratum with the same lattice Veech group.
Our methods can be applied to obtain obstructions of lattices being realized as Veech groups in certain strata; in particular, we show that the square torus is the only translation surface in any minimal stratum whose Veech group is all of SL2Z.
Original languageEnglish
PublisherarXiv
Pages1-28
DOIs
Publication statusPublished - 29 Nov 2021

Keywords

  • translation surfaces
  • Veech group

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