Rapid Decay Property with Examples in Crystallographic and Discrete Heisenberg Groups

Abstract

Rapid Decay Property (Property (RD)) provides estimates for the operator norm of those functions (in the left-regular representation) in terms of the Sobolev norm. We provide the examples of length function and related some results. We study the stability of property (RD) with respect to some constructions of groups. We show the some lemmas related for Heisenberg groups. Finally, we show that the following groups have property (RD): • The Crystallographic groups. • The Discrete Heisenberg group.