Centroids and the Rapid Decay property in mapping class groups
  We study a notion of a Lipschitz, permutation-invariant "centroid" for triples of points in mapping class groups MCG(S), which satisfies a certain polynomial growth bound. A consequence (via work of Drutu-Sapir or Chatterji-Ruane) is the Rapid Decay Property for MCG(S). 
Jason A. Behrstock and Yair N. Minsky