Finiteness results for flat surfaces: large cusps and short geodesics

TL;DR

This paper establishes finiteness results for flat surfaces with certain Veech group properties, providing new restrictions on their structure and classification within fixed genus and cusp bounds.

Contribution

It proves finiteness of affine classes with bounded cusps, and shows non-elementary Veech groups appear finitely often and have finite index in their normalizers.

Findings

Finiteness of affine classes with bounded hyperbolic cusps.

Non-elementary Veech groups appear finitely often in a fixed stratum.

Quotients by non-lattice Veech groups contain arbitrarily large embedded disks.

Abstract

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.