On the degree of rapid decay

Abstract

A finitely generated group $\G$ equipped with a word-length is said to satisfy property RD if there are $C,s\ge 0$ such that, for all non-negative integers $n$, we have $\Vert a\Vert \le C(1+n{)}^s\Vert a{\Vert }_2$ 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.

arXiv comment6 pages, final version