Pseudo-Anosovs are exponentially generic in mapping class groups

Choi, Inhyeok

doi:10.2140/gt.2024.28.1923

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Pseudo-Anosovs are exponentially generic in mapping class groups

Inhyeok Choi

Geometry & Topology 28 (2024) 1923–1955

DOI: 10.2140/gt.2024.28.1923

Abstract

Given a finite generating set $S$ , let us endow the mapping class group of a closed hyperbolic surface with the word metric for $S$ . We discuss the following question: does the proportion of non-pseudo-Anosov mapping classes in the ball of radius $R$ converge to $0$ as $R$ tends to infinity? We show that any finite subset $S^{'}$ of the mapping class group is contained in a finite generating set $S$ such that this proportion decays exponentially. Our strategy applies to weakly hyperbolic groups and does not refer to the automatic structure of the group.

Keywords

mapping class group, pseudo-Anosov, random walk

Mathematical Subject Classification

Primary: 20F67, 30F60, 57K20, 57M60, 60G50

References

Publication

Received: 9 November 2021

Revised: 18 August 2022

Accepted: 24 September 2022

Published: 18 July 2024

Proposed: Mladen Bestvina

Seconded: Benson Farb, David Fisher

Authors

Inhyeok Choi
June E Huh Center for Mathematical Challenges KIAS Seoul South Korea

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Volume 28, issue 4 (2024)