Pseudo-Anosovs are exponentially generic in mapping class groups

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 $R$ decrease to 0 as $R$ increases? 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 decreases exponentially. Our strategy applies to weakly hyperbolic groups and does not refer to the automatic structure of the group.

Expanded the introduction and elaborated several details. 33 pages and 6 figures. The final version of this paper will appear in Geometry and Topology

References in corpus (2)

• Exponential bounds for random walks on hyperbolic spaces without moment conditions
• Central limit theorem and geodesic tracking on hyperbolic spaces and Teichmüller spaces

Cited by in corpus (2)

• Counting pseudo-Anosovs as weakly contracting isometries
• Random walks and contracting elements I: Deviation inequality and Limit laws