Constructing lattice surfaces with prescribed Veech groups: an algorithm

Slade Sanderson*

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review


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 SL 2Z .

Original languageEnglish
Article number83
Number of pages37
JournalGeometriae Dedicata
Issue number5
Publication statusPublished - Oct 2023


  • Fuchsian groups
  • Riemann surfaces
  • Translation surfaces
  • Veech groups


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