The rapid decay property for pairs of discrete groups

TL;DR

This paper extends the rapid decay property concept from groups to pairs of groups, exploring its implications for K-theory, spectral injections, and random walk probabilities.

Contribution

It introduces a generalized notion of rapid decay for group pairs and investigates its effects on K-theory isomorphisms and spectral properties.

Findings

Establishes isomorphisms in K-theory for group pairs with the generalized rapid decay property.

Analyzes relatively spectral injections in reduced group C*-algebras.

Provides bounds on return probabilities of symmetric random walks to subgroup H.

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 $C^*$-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 $G$.