The Rapid Decay property and centroids in groups
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.
Cite
@article{arxiv.1405.0757,
title = {The Rapid Decay property and centroids in groups},
author = {Mark Sapir},
journal= {arXiv preprint arXiv:1405.0757},
year = {2015}
}
Comments
27 pages; v2: Section 2 corrected, added a reference to Olshansii's preprint arXiv:1406.0336 in Section 4. v3: Section 3.6 on graph products of groups is added. v3: Small correction in the proof of Theorem 2.3, estimate slightly improved