n the article we settle down the problem of permanence of property RD under group extensions. We show that if $1\to N\to G\to Q\to 1$ is a short exact sequence of compactly generated groups such that $Q$ has property RD, and $N$ has property RD with respect to the restriction of a word-length on $G$, then $G$ has property RD.
We also generalize the result of Ji and Schweitzer stating that locally compact groups with property RD are unimodular. Namely, we show that any automorphism of a locally compact group with property RD which distorts distances subexponentially, preserves the Haar measure.
Lukasz Garncarek. "Property of rapid decay for extensions of compactly generated groups." Publ. Mat. 59 (2) 301 - 312, 2015.