Quasisymmetric maps of boundaries of amenable hyperbolic groups Tullia Dymarz 20F6530C6553C20quasi-isometryquasisymmetric mapnegative curvature 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-Carath\'eodory metric, then any quasisymmetric map of $Y$ is actually bilipschitz. The key observation is that $Y$ is the \emph{parabolic visual boundary} of a \emph{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 $N=\mathbb{R}^n$. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5214 10.1512/iumj.2014.63.5214 en Indiana Univ. Math. J. 63 (2014) 329 - 343 state-of-the-art mathematics http://iumj.org/access/