Euler characteristics of SL(2,Z)-orbit graphs of square-tiled surfaces

Abstract: Square-tiled surfaces (aptly named) are surfaces obtained by gluing together a collection of unit squares along their sides (the square torus being the simplest example). These surfaces are special cases of translation surfaces, whose moduli space carries a natural action of the group $SL(2,\mathbb{R})$. The famous works of Eskin–Mirzakhani and Eskin–Mirzakhani–Mohammadi were concerned with understanding the orbits of this action.

This $SL(2,\mathbb{R})$-action restricts to an action of $SL(2,\mathbb{Z})$ on square-tiled surfaces, and the orbits under this restricted action can be transformed into finite regular graphs. It is a long-standing conjecture of McMullen that a specific family of these orbit graphs in genus two forms a family of expander graphs. Providing indirect evidence for this conjecture, I will describe joint work with Carlos Matheus in which we prove that the absolute values of the Euler characteristics of the graphs in this family go to infinity (a requirement for any expander family). To do this, we are required to count a variety of objects, including integer points on algebraic hypersurfaces, pseudo-Anosov homeomorphisms of fixed dilatation, and orbifold points on certain algebraic curves in moduli space.

algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry

Audience: researchers in the topic

Geometry and topology online

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The talks start at 13:30. Talks are typically fifty minutes long, with ten minutes for questions.