Stable Type of the Mapping Class Group

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 $III_λ$ for some $λ>0$. 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 $PMF$.

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