JournalscmhVol. 85, No. 2pp. 313–336

Finiteness results for flat surfaces: large cusps and short geodesics

  • John Smillie

    Cornell University, Ithaca, USA
  • Barak Weiss

    Tel Aviv University, Israel
Finiteness results for flat surfaces: large cusps and short geodesics cover
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Abstract

For fixed g and T we show the finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech group contains a cusp of hyperbolic co-area less than T. We obtain new restrictions on Veech groups: any non-elementary Fuchsian group can appear only finitely many times in a fixed stratum; any non-elementary Veech group is of finite index in its normalizer; and the quotient of ℍ by a non-lattice Veech group admits arbitrarily large embedded disks. A key ingredient of the proof is the finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech group contains a hyperbolic element with eigenvalue less than T.

Cite this article

John Smillie, Barak Weiss, Finiteness results for flat surfaces: large cusps and short geodesics. Comment. Math. Helv. 85 (2010), no. 2, pp. 313–336

DOI 10.4171/CMH/197