Asymptotic geometric and probabilistic properties of Teichmüller space

Azemar, Aitor (2024) Asymptotic geometric and probabilistic properties of Teichmüller space. PhD thesis, University of Glasgow.

Abstract

Teichmüller spaces play a pivotal role in the study of dynamics, geometric group theory and conformal geometry. In this thesis we study several geometric and probabilistic aspects of these spaces.

We begin by showing that, while not hyperbolic, Teichmüller space with the Teichmüller distance is statistically hyperbolic with respect to harmonic measures generated by nonelementary measures with finite first moment.

Points in the Teichmüller space of a surface 𝘚 can be interpreted as conjugacy classes of discrete faithful representations of the fundamental group of 𝘚 on the group of isometries of the hyperbolic plane, PSL(2,ℝ). For a given measure on the fundamental group of S, this characterization gives us associated measures on uniform lattices on PSL(2,ℝ). It is a long standing conjecture that the harmonic measures associated to these random walks have dimension strictly smaller than one whenever the measure is admissible and has finite first moment. In this thesis we prove that the conjecture is true outside of a compact subset of the Teichmüller space. Furthermore, we give some sharp bounds for the growth of the drift of the associated random walks in terms of the Teichmüller distance. One key argument is an adaptation of Gouëzel's pivoting techniques to actions of a fixed group on a sequence of hyperbolic metric spaces.

Two commonly studied compactifications of Teichmüller spaces of finite type surfaces are the Gardiner-Masur compactification and the Teichmüller compactification. We finish by showing that these two compactifications are related, proving that the former is finer than the latter. This allows us to prove, among other results, that the Gardiner-Masur compactification is path connected and that its Busemann points are not dense. We also determine for which surfaces the two compactifications are isomorphic, and we show that some horocycles diverge in the Teichmüller compactification based at some point. As an ingredient in one of the proofs we show that the extremal length is not 𝐶² along some paths that are smooth with respect to the piecewise linear structure on measured foliations.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics
• Supervisor's Name: Gadre, Dr. Vaibhav, Brendle, Professor Tara and Fortier-Bourque, Dr. Maxime
• Date of Award: 2024
• Depositing User: Theses Team
• Unique ID: glathesis:2024-84802
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 13 Jan 2025 10:32
• Last Modified: 13 Jan 2025 10:33
• Thesis DOI: 10.5525/gla.thesis.84802
• URI: https://theses.gla.ac.uk/id/eprint/84802