Large scale geometry of negatively curved $\mathbb{R}^n \rtimes \mathbb{R}$

Abstract

We classify all negatively curved ${ℝ}^n⋊ℝ$ up to quasi-isometry. We show that all quasi-isometries between such manifolds (except when they are bilipschitz to the real hyperbolic spaces) are almost similarities. We prove these results by studying the quasisymmetric maps on the ideal boundary of these manifolds.