Property of rapid decay for extensions of compactly generated groups

Abstract

ABSTRACT. 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.

Introduction to the Rapid Decay property

“…From the definition of Rapid Decay property it is clear that the property is inherited by subgroups with the induced length, and hence having an amenable subgroup of super-polynomial growth is an obstruction to the Rapide Decay property, showing that for instance SL 3 (Z) does not have that property. The above example also shows that in general the Rapid Decay property is not stable under taking extensions, however, in a short exact sequence of finitely generated groups {e} → Z → G → Q → {e} then G has the Rapid Decay property if and only if Q has the Rapid Decay property and Z has the Rapid Decay property for the induced length from G, see [24] for the general statement, and Remark 4.3 for a discussion on the Rapid Decay property and length functions.…”

“…then G has the Rapid Decay property if and only if Q has the Rapid Decay property and Z has the Rapid Decay property for the induced length from G, see [24] for the general statement, and Remark 4.3 for a discussion on the Rapid Decay property and length functions.…”

Mini-course: property of rapid decay

“…Recently, the following strengthening of condition (5) in Theorem 3.1, omitting the assumption of polynomial distortion, was proved in [6] (actually, in the context of topological compactly generated groups). Theorem 3.2.…”

Real hyperbolic hyperplane complements in the complex hyperbolic plane

“…, P n do. So it remains to check the RDP for the fundamental groups of the ends of M \ S. But the ends of M \ S are either infranil manifolds or circle bundles over components of S. In the first case, the RDP was established by Jolissaint in [Jol90], and in the second case it is established by Garncarek in [Gar15] (also, one could combine [Jol90] with [Nos92]). Finally, Lafforgue [Laf02] proved that the fundamental group of any complete Riemannian manifold equipped with a nonpositively curved A-regular metric which satisfies the RDP must also satisfy the Baum-Connes conjecture.…”