On the degree of rapid decay

Abstract

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