Non-planarity of SL(2,Z)-orbits of origamis in genus two

Abstract:

Origamis (also known as square-tiled surfaces) arise naturally in a variety of settings in low-dimensional topology. They can be thought of as surfaces obtained by gluing the sides of a collection of unit squares. As such, they generalise the torus which can be obtained by gluing the sides of a single square. An origami is said to be primitive if it is not a cover of a lower genus origami.

In this talk, I will describe how one can define an action of the matrix group \(\mathrm{SL}(2,\mathbb{Z})\) on primitive origamis. In genus two (with one singularity), the orbits of this action were classified by Hubert-Lelièvre and McMullen. By considering a generating set of size two for \(\mathrm{SL}(2,\mathbb{Z})\), we can turn these orbits into an infinite family of four-valent graphs. For a specific generating set, I will explain how all but two of these graphs are non-planar. I will also discuss why this gives indirect evidence for McMullen's conjecture that these graphs form a family of expanders.

This is joint work with Carlos Matheus.

algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry

Audience: researchers in the topic

( paper | slides )

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Geometry and topology online

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The talks start five minutes after the hour. Talks are typically 55 minutes long, including time for questions.

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