Finiteness results for flat surfaces: large cusps and short geodesics

Details Finiteness results for flat surfaces: large cusps and short geodesics Finiteness results for flat surfaces: large cusps and short geodesics

2008-02-07

arxiv.org/abs/0802.0919v1

Mathematics

Dynamical Systems

Scientific paper

For fixed g and T we show that finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech groups contain a cusp of hyperbolic co-area less than T. We obtain new restrictions on Veech groups: we show that any non-elementary Veech group can appear only finitely many times in a fixed stratum, that any non-elementary Veech group is of finite index in its normalizer, and that the quotient of the upper half plane by a non-lattice Veech group contains arbitrarily large embedded disks. These are proved using 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.

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