Finiteness results for flat surfaces: large cusps and short geodesics
John Smillie
Cornell University, Ithaca, USABarak Weiss
Tel Aviv University, Israel

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