Hyperbolic octagons and Teichmüller space in genus 2

Abstract

In this paper we study different models of the Teichmüller space of compact hyperbolic surfaces with special emphasis on their construction by geodesic octagons. Such surfaces have become increasingly interesting due to their applications to the physical behavior of chaotic quantum systems. They are, on the one hand, complex enough to show the relevant features, and on the other hand, they possess a simple mathematical structure allowing practical implementations. We give a new description of Teichmüller space and the mapping class group in terms of geometric data of the octagons. This provides modellings based on the vertices and also on the generators of the associated isometry group. In addition, we explicitly connect our approach with other models currently used in the literature (Helling matrices, Fenchel–Nielsen parameters). The result of the paper may be considered as a computational tool kit to work with a specific model and to relate it with the others.