The Thurston spine and equivariant cell decompositions in genus 2

Abstract

The Thurston spine of the genus 2 Teichmüller space of closed compact surfaces is obtained. In particular, we identify all mapping class group-orbits of filling sets of systoles in genus 2. Thereby, surprisingly rich cases already appear in this simplest case which is genus 2. This is contrasted with a cell decomposition of a bordification of Teichmüller space obtained computationally using Rivin's angle coordinates, on which the cells are labelled by Delaunay graphs closely related to our systolic graphs.