Large scale geometry of negatively curved ℝn ⋊ ℝ

Xie, Xiangdong

doi:10.2140/gt.2014.18.831

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

Xiangdong Xie

Geometry & Topology 18 (2014) 831–872

DOI: 10.2140/gt.2014.18.831

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

References

Publication

Received: 20 July 2012

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/

Volume 18, issue 2 (2014)