Volume 18, issue 2 (2014)

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

Xiangdong Xie

Geometry & Topology 18 (2014) 831–872
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.

Keywords
quasiisometry, quasisymmetric map, negatively curved solvable Lie groups
Mathematical Subject Classification 2010
Primary: 20F65, 30C65
Secondary: 53C20