Centroids and the Rapid Decay property in mapping class groups

TL;DR

This paper introduces a Lipschitz, permutation-invariant centroid concept for triples in mapping class groups, leading to the proof of the Rapid Decay Property for these groups, which has implications in geometric group theory.

Contribution

It defines a new centroid notion satisfying polynomial growth bounds and uses it to establish the Rapid Decay Property for mapping class groups.

Findings

Established the Rapid Decay Property for MCG(S)

Introduced a new centroid concept with polynomial growth bounds

Connected centroid properties to group analytic properties

Abstract

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).