Algebraic & Geometric Topology

Wild translation surfaces and infinite genus

Anja Randecker

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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.

Article information

Algebr. Geom. Topol., Volume 18, Number 5 (2018), 2661-2699.

Received: 8 February 2017
Revised: 5 October 2017
Accepted: 11 March 2018
First available in Project Euclid: 30 August 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C10: $G$-structures
Secondary: 37D50: Hyperbolic systems with singularities (billiards, etc.) 37E35: Flows on surfaces 57M50: Geometric structures on low-dimensional manifolds

translation surfaces wild singularities infinite topological type


Randecker, Anja. Wild translation surfaces and infinite genus. Algebr. Geom. Topol. 18 (2018), no. 5, 2661--2699. doi:10.2140/agt.2018.18.2661.

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