Finiteness results for flat surfaces: large cusps and short geodesics

  • John Smillie

    Cornell University, Ithaca, USA
  • Barak Weiss

    Tel Aviv University, Israel

Abstract

For fixed and we show the finiteness of the set of affine equivalence classes of flat surfaces of genus whose Veech group contains a cusp of hyperbolic co-area less than . 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 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 whose Veech group contains a hyperbolic element with eigenvalue less than .

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