Pseudo-Anosovs are exponentially generic in mapping class groups

Choi, Inhyeok¹

¹ June E Huh Center for Mathematical Challenges, KIAS, Seoul, South Korea

Geometry & topology, Tome 28 (2024) no. 4, pp. 1923-1955.

Résumé

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 $S^{\prime }$ 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.

Détail

DOI : 10.2140/gt.2024.28.1923

Keywords: mapping class group, pseudo-Anosov, random walk

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