Extremal Hyperbolic Surfaces and the Selberg Trace Formula

The Selberg trace formula provides a link between the length spectrum and the Laplacian spectrum of a hyperbolic surface. I will speak about a joint project with Maxime Fortier Bourque in which we are using this formula to probe extremal problems in hyperbolic geometry. These are questions of the form: what is the hyperbolic surface of a given genus with the largest kissing number or the largest spectral gap? Concretely, I will explain the general principle of our method, which is inspired by ideas from the world of Euclidean sphere packings. Moreover, I will explain why the Klein quartic, the most symmetric Riemann surface in genus 3, solves one of our extremal problems.

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