Random Walks on the Mapping Class Group

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Anvari, Leila | 2024

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 58130 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Safdari, Mohammad; Nassiri, Maysam; Talebi, Aminossadat
Abstract:
Let Σ be a connected orientable surface of finite type. Due to Nielsen and Thurston’s classification, it is known that any element of the mapping class group of Σ is periodic, reducible, or (exclusively) pseudo-Anosov. Thus it is natural to form a random walk on the mapping class group with a step distribution and ask about the asymptotic distribution of the location of the random walk on each of the mentioned classes. Joseph Maher has proved that if the group generated by the support of is a non-elementary subgroup of the mapping class group, then the random walk will encounter a pseudo-Anosov element with asymptotic probability one. Indeed, this result not only concerns random walks on the mapping class group, but on many of its subgroups, including the Torelli group. In this thesis, after introducing the preliminaries, we follow Maher’s proof of the mentioned theorem
Keywords:
Teichmuller Space Compactification ; Gromov Theorem ; Random Walk ; Curve Complex ; Mapping Class Group ; Pseudo-Anosov Elements