The rapid decay property for pairs of discrete groups

Abstract

We generalize the notion of rapid decay property for a group $G$ to pairs of groups $(G,H)$ where $H$ is a finitely generated subgroup of $G$, where typically the subgroup $H$ does not have rapid decay. We deduce some isomorphisms in $K$-theory, and investigate relatively spectral injections in the reduced group $C^{\ast }$-algebra. Rapid decay property for the pair $(G,H)$ also gives a lower bound for the probability of return to $H$ of symmetric random walks on $G$.

25 pages, 0 figures. What was previously Theorem 3.1 missed the assumption polynomial growth and had a gap in the proof. It is now Corollary 3.1