Geometry & Topology

Coarse differentiation and quasi-isometries of a class of solvable Lie groups I

Irine Peng

Full-text: Open access

Abstract

This is the first of two consecutive papers that aim to understand quasi-isometries of a class of unimodular split solvable Lie groups. In the present paper, we show that locally (in a coarse sense), a quasi-isometry between two groups in this class is close to a map that respects their group structures. In the following paper we will use this result to show quasi-isometric rigidity.

Article information

Source
Geom. Topol., Volume 15, Number 4 (2011), 1883-1925.

Dates
Received: 13 April 2009
Revised: 3 August 2011
Accepted: 3 August 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732357

Digital Object Identifier
doi:10.2140/gt.2011.15.1883

Mathematical Reviews number (MathSciNet)
MR2860983

Zentralblatt MATH identifier
1235.51019

Subjects
Primary: 51F99: None of the above, but in this section
Secondary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]

Keywords
quasi-isometry solvable group rigidity

Citation

Peng, Irine. Coarse differentiation and quasi-isometries of a class of solvable Lie groups I. Geom. Topol. 15 (2011), no. 4, 1883--1925. doi:10.2140/gt.2011.15.1883. https://projecteuclid.org/euclid.gt/1513732357


Export citation

References

  • L Auslander, An exposition of the structure of solvmanifolds, I: Algebraic theory, Bull. Amer. Math. Soc. 79 (1973) 227–261
  • M R Bridson, S M Gersten, The optimal isoperimetric inequality for torus bundles over the circle, Quart. J. Math. Oxford Ser. $(2)$ 47 (1996) 1–23
  • T Dymarz, Large scale geometry of certain solvable groups, Geom. Funct. Anal. 19 (2010) 1650–1687
  • T Dymarz, I Peng, Bilipschitz maps of boundaries of certain negatively curved homogeneous spaces, Geom. Dedicata 152 (2011) 129–145
  • A Dyubina, Instability of the virtual solvability and the property of being virtually torsion-free for quasi-isometric groups, Internat. Math. Res. Notices (2000) 1097–1101
  • A Eskin, D Fisher, K Whyte, Quasi-isometries and rigidity of solvable groups, Pure Appl. Math. Q. 3 (2007) 927–947
  • A Eskin, D Fisher, K Whyte, Coarse differentiation of quasi-isometries I: Spaces not quasi-isometric to Cayley graphs
  • A Eskin, D Fisher, K Whyte, Coarse differentiation of quasi-isometries II: Rigidity for Sol and Lamplighter groups
  • Y Guivarc'h, Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire, from: “Conference on Random Walks (Kleebach, 1979)”, Astérisque 74, Soc. Math. France, Paris (1980) 47–98, 3 In French
  • D V Osin, Exponential radicals of solvable Lie groups, J. Algebra 248 (2002) 790–805
  • I Peng, Coarse differentiation and quasi-isometries of a class of solvable Lie groups II, Geom. Topol. 15 (2011)