SL(2,Z)-invariants of single-cylinder square-tiled surfaces

Luke Jeffreys (University of Bristol)

Monday 9th December 16:00-17:00 Maths 311B

Abstract

Translation surfaces are surfaces obtained by identifying opposite sides of Euclidean polygons (think of gluing the sides of a square to get a torus). The space of translation surfaces carries a natural and well-studied action by SL(2,R). When all the polygons used to construct the surface are squares, we say that the surface is square-tiled. On the set of square-tiled surfaces, the action of SL(2,R) restricts to an action of SL(2,Z). The classification of SL(2,Z)-orbits of square-tiled surfaces has only been carried out in a few cases and is wide open in general.

In this talk, in an attempt to understand their SL(2,Z)-orbits, I will discuss work calculating certain SL(2,Z)-invariants (monodromy and spin parity) of a family of (single-cylinder) square-tiled surfaces recently constructed by Aougab-Menasco-Nieland. I will also present some computational evidence suggesting a connection between monodromy groups of single-cylinder square-tiled surfaces and finite simple groups. This is joint work with Tarik Aougab, Adam Friedman-Brown, and Jiajie Ma.

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