Quasisymmetric maps of boundaries of amenable hyperbolic groups

Abstract

In this paper we show that if Y = N ×Qm is a metric space where N is a Carnot group endowed with the Carnot-Caratheodory metric then any quasisymmetric map of Y is actually bilipschitz. The key observation is that Y is the parabolic visual boundary of a mixed type locally compact amenable hyperbolic group. The same results also hold for a larger class of nilpotent Lie groups N . As part of the proof we also obtain partial quasi-isometric rigidity results for mixed type locally compact amenable hyperbolic groups. Finally we prove a rigidity result for uniform subgroups of bilipschitz maps of Y in the case of N = Rn.