Property of rapid decay for extensions of compactly generated groups

In the article we settle down the problem of permanence of property RD under group extensions. We show that if 1→ N→ G→ Q→ 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.