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

Various commensurability relations in Hecke pairs and property (RD)