Wild translation surfaces and infinite genus

TL;DR

This paper investigates the relationship between wild singularities and the topology of translation surfaces, demonstrating that certain wild singularities imply the surface has infinite genus, especially in parabolic or finite cases.

Contribution

It establishes conditions under which wild singularities lead to infinite genus in translation surfaces, extending classical relations to more general cases.

Findings

Wild singularities can imply infinite genus.

Parabolic or finite translation surfaces with wild singularities have infinite genus.

Conditions for wild singularities to influence surface topology.

Abstract

The Gauss-Bonnet formula for classical translation surfaces relates the cone angle of the singularities (geometry) to the genus of the surface (topology). When considering more general translation surfaces, we observe so-called wild singularities for which the notion of cone angle is not applicable any more. We study whether there still exist relations between the geometry and the topology for translation surfaces with wild singularities. By considering short saddle connections, we determine under which conditions the existence of a wild singularity implies infinite genus. We apply this to show that parabolic or essentially finite translation surfaces with wild singularities have infinite genus.