Overview
This research focuses on the Thurston spine within the context of the genus 2 Teichmüller space, specifically for closed compact surfaces. A primary outcome is the explicit acquisition of this Thurston spine. Concurrently, the study identifies all mapping class group-orbits pertaining to filling sets of systoles, specifically within genus 2. The authors note that this particular case, genus 2, unexpectedly reveals a rich array of scenarios.
Research Context
The Teichmüller space is a fundamental concept in mathematics, particularly in the study of Riemann surfaces and hyperbolic geometry. The Thurston spine is a significant structural component within this space. This work specifically targets the genus 2 instance of Teichmüller space, which represents the simplest case where complex phenomena related to these structures begin to manifest. The investigation into mapping class group-orbits of filling sets of systoles contributes to a deeper understanding of the geometric and topological properties of these surfaces.
Approach
The core approach involved obtaining the Thurston spine for the genus 2 Teichmüller space. A key aspect of this process was the identification of all mapping class group-orbits of filling sets of systoles specifically within genus 2. This identification addresses a particular aspect of the structure of these spaces. The work also includes a comparison of these findings with a distinct computational method. This contrast involves a cell decomposition of a bordification of Teichmüller space, which was derived computationally. This computational decomposition utilized Rivin's angle coordinates and featured cells labeled by Delaunay graphs. These Delaunay graphs are noted to be closely related to the systolic graphs developed in the primary analysis.
Findings
- The Thurston spine of the genus 2 Teichmüller space of closed compact surfaces has been obtained.
- All mapping class group-orbits of filling sets of systoles in genus 2 have been identified.
- The genus 2 case, despite being the simplest, exhibits surprisingly rich characteristics concerning these structures.
- The results are contrasted with a computational cell decomposition of a bordification of Teichmüller space. This decomposition was obtained using Rivin's angle coordinates.
- The cells in the computational decomposition are labelled by Delaunay graphs, which are indicated to be closely related to the systolic graphs derived in the main part of the study.
Why This Matters
The explicit acquisition of the Thurston spine for genus 2, along with the detailed identification of mapping class group-orbits of filling sets of systoles, provides foundational insights into the structure of Teichmüller space at its simplest non-trivial level. This contributes to the understanding of the geometric and topological properties of closed compact surfaces of genus 2. The contrast with computational methods using Rivin's angle coordinates and Delaunay graphs highlights connections between different mathematical approaches to characterizing these complex spaces.