#### Volume 18, issue 2 (2014)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
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
##### Publication
Revised: 5 May 2013
Accepted: 28 September 2013
Published: 20 March 2014
Proposed: Dmitri Burago
Seconded: Benson Farb, John Lott
##### Authors
 Xiangdong Xie Department of Mathematics and Statistics Bowling Green State University Bowling Green, OH 43403 USA http://personal.bgsu.edu/~xiex/