Stable Type of the Mapping Class Group

TL;DR

This paper demonstrates that the action of the mapping class group on the space of projective measured foliations exhibits stable type III_λ, using dynamics of Teichmüller geodesic flow and generalized Bowen criteria.

Contribution

It introduces a new approach to classify the stable type of the mapping class group action via Teichmüller flow and Patterson-Sullivan type results.

Findings

Mapping class group action has stable type III_λ for some λ>0

Generalization of Bowen's criterion for stable ratio set

Establishment of Patterson-Sullivan type results for Thurston measure

Abstract

We use dynamics of the Teichm$\ddot{\mathrm{u}}$ller geodesic flow to show that the action of the mapping class group on the space of projective measured foliations has stable type $III_{\lambda}$ for some $\lambda>0$. We do this by generalizing a criterion due to Bowen for a number to be in the stable ratio set, and proving some Patterson-Sullivan type results for the Thurston measure on $PMF$.