On the degree of rapid decay

TL;DR

This paper investigates the property RD in finitely generated groups, establishing a lower bound of 1/2 for the degree s in the case of infinite groups, contributing to understanding the decay properties of group algebras.

Contribution

It proves that for infinite finitely generated groups, the degree s in property RD cannot be less than 1/2, providing a fundamental lower bound.

Findings

The degree s in property RD is at least 1/2 for infinite groups.

Establishes a quantitative lower bound on decay rates in group algebras.

Advances theoretical understanding of decay properties in geometric group theory.

Abstract

A finitely generated group $\G$ equipped with a word-length is said to satisfy property RD if there are $C, s\geq 0$ such that, for all non-negative integers $n$, we have $\|a\|\leq C (1+n)^s \|a\|_2$ whenever $a\in\C\G$ is supported on elements of length at most $n$. We show that, for infinite $\G$, the degree $s$ is at least 1/2.