  In this paper we show that if $Y=N \times \mathbb{Q}_m$ 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 $N= \mathbb{R}^n$.

Group Theory;Metric Geometry