Geometry & Topology

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

Xiangdong Xie

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We classify all negatively curved n 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.

Article information

Geom. Topol., Volume 18, Number 2 (2014), 831-872.

Received: 20 July 2012
Revised: 5 May 2013
Accepted: 28 September 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 30C65: Quasiconformal mappings in $R^n$ , other generalizations
Secondary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]

quasiisometry quasisymmetric map negatively curved solvable Lie groups


Xie, Xiangdong. Large scale geometry of negatively curved $\mathbb{R}^n \rtimes \mathbb{R}$. Geom. Topol. 18 (2014), no. 2, 831--872. doi:10.2140/gt.2014.18.831.

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