The Farrell–Jones Conjecture for mapping class groups

Abstract

We prove the Farrell–Jones Conjecture for mapping class groups. The proof uses the Masur–Minsky theory of the large scale geometry of mapping class groups and the geometry of the thick part of Teichmuller space. The proof is presented in an axiomatic setup, extending the projection axioms in Bestvina et al. (Publ Math Inst Hautes Etudes Sci 122:1–64, 2015). More specifically, we prove that the action of $${{\,\mathrm{Mod(\Sigma )}\,}}$$ on the Thurston compactification of Teichmuller space is finitely $$\mathcal F$$ -amenable for the family $${\mathcal {F}}$$ consisting of virtual point stabilizers.