Various commensurability relations in Hecke pairs and property (RD)

TL;DR

This paper investigates how property (RD) behaves under various subgroup relations in Hecke pairs, providing new criteria and examples, including cases involving normal subgroups, quotients, and group extensions.

Contribution

It introduces new results on property (RD) for Hecke pairs under different commensurability relations and constructs new classes of groups with property (RD).

Findings

Hecke pairs with normal subgroups have (RD) iff their quotients do.

Introduces a class of groups with all subgroups almost normal, all having property (RD).

Analyzes property (RD) in Hecke pairs from group extensions.

Abstract

In this paper we continue our study of property (RD) for Hecke pairs initiated in (V. Shirbisheh, Property (RD) for Hecke pairs. Mathematical Physics, Analysis and Geometry, vol. 15, no. 2, (2012), 173--192). We study the behavior of property (RD) under different commensurability relations of subgroups in Hecke pairs. As an application, we prove that if H is a normal subgroup of a group G and K is a subgroup of G commensurable to H, then the Hecke pair (G,K) has (RD) if and only if the quotient group G/H has (RD). This is used to investigate an infinite number of non-elementary examples of Hecke pairs with property (RD). In particular, we introduce a class of groups whose all subgroups are almost normal and all such Hecke pairs have property (RD). We also discuss property (RD) of certain Hecke pairs arising from group extensions.