Wild translation surfaces and infinite genus

Randecker, Anja

doi:10.2140/agt.2018.18.2661

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Wild translation surfaces and infinite genus

Anja Randecker

Algebraic & Geometric Topology 18 (2018) 2661–2699

DOI: 10.2140/agt.2018.18.2661

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.

Keywords

translation surfaces, wild singularities, infinite topological type

Mathematical Subject Classification 2010

Primary: 53C10

Secondary: 37D50, 37E35, 57M50

References

Publication

Received: 8 February 2017

Revised: 5 October 2017

Accepted: 11 March 2018

Published: 22 August 2018

Authors

Anja Randecker
Department of Mathematics University of Toronto Toronto, ON Canada
http://www.math.toronto.edu/anja/

Volume 18, issue 5 (2018)