Mini-course: property of rapid decay

TL;DR

This mini-course notes explore the property of rapid decay (RD) in groups, discussing its definitions, permanence properties, and classes of groups that possess RD, with insights into related concepts like the centroid property.

Contribution

It provides an accessible overview of RD, including definitions, key properties, and proofs for specific classes of groups, filling gaps in existing lecture materials.

Findings

Certain classes of groups have the property of rapid decay.

The centroid and relative centroid properties are important in establishing RD.

The notes clarify the definitions and permanence properties of RD.

Abstract

These are the notes for the mini-course on the property of rapid decay given at the Erwin Schr\"odinger Institute in Vienna between March 14th and 18th, 2016, as a part of the Measured Group Theory program. They include parts which were omitted during the lectures, and omit some parts which were included in the lectures (and now I think that they shouldn't have been). We explore the definitions of property RD and its permanence properties, discuss the centroid property and relative centroid property defined by Mark Sapir, and give proofs that certain classes of groups have RD.