Geometry & Topology

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

Xiangdong Xie

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732759

Digital Object Identifier
doi:10.2140/gt.2014.18.831

Mathematical Reviews number (MathSciNet)
MR3180486

Zentralblatt MATH identifier
1344.53036

Subjects
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]

Keywords
quasiisometry quasisymmetric map negatively curved solvable Lie groups

Citation

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. https://projecteuclid.org/euclid.gt/1513732759


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