The Rapid Decay property and centroids in groups

TL;DR

This paper surveys methods for establishing the Rapid Decay property in groups, introduces a centroid property linked to group actions on metric spaces, and demonstrates its implications for various classes of groups.

Contribution

It introduces a generalized centroid property that implies RD and connects it with known properties, expanding understanding of groups with RD.

Findings

Several properties imply RD and the centroid property.

Uniform lattices, Artin groups of large type, and mapping class groups have the centroid property.

A simple non-amenability-like property is derived from RD.

Abstract

This is a survey of methods of proving or disproving the Rapid Decay property in groups. We present a centroid property of group actions on metric spaces. That property is a generalized (and corrected) version of the property (**)-relative hyperbolicity" from our paper with Cornelia Drutu, math/0405500, and implies the Rapid Decay (RD) property. We show that several properties which are known to imply RD also imply the centroid property. Thus uniform lattices in many semi-simple Lie groups, Artin groups of large type and the mapping class groups have the centroid property. We also present a simple "non-amenability-like" property that follows from RD, and give an easy example of a group without RD and without any amenable subgroup with superpolynomial growth, some misprints in other sections are corrected.