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.