Various commensurability relations in Hecke pairs and property (RD)

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.