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 $C, s\geq 0$ 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.Comment: 6 pages, final versio