Systoles and Coxeter groups

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:Olivier Mathieu(法国国家科学研究院)
:2024-04-02 10:30

报告人:Olivier Mathieu法国国家科学研究院




A systole of a closed hyperbolic surface is a closed geodesic of shortest length. A generic surface of genus g ≥ 2 has a unique systole which cuts it into one or two noncontratible pieces. However, there are closed hyperbolic surfaces of genus g, whose set of systoles fills the surface, i.e. it cuts the surface into polygons. In general, it is easy to find surfaces with a large number of systoles which fill. A classical example is the Bolsa surface of genus 2 with 12 systoles.

Here we are interested to find surfaces for which the set of systoles is filling while being small. In fact, our approach provides surfaces with significatively less systoles than the previously known examples. Then we will briefly outline a consequence in Teichmüller theory.

Our construction is based on the theory of Coxeter groups, especially the arithmetic properties of the Tits representation. From a geometric viewpoint, it involves hyperbolic tessellations. Although this work mixes hyperbolic geometry and representation theory, our presentation will be quite elementary.


Professor Olivier Mathieu is a senior CNRS researcher in France. He got the Grant Sloan fellowship (1991-94), the Grand prix IBM-France (1992). He was an invited speaker at the ICM (Kyoto 1990), plenary speaker of the first joint Brazil-France Conference, 2019 (Rio), and gave Jose Adem Memorial Lectures Series, 2019 (Mexico).