Quasi-isometries and rigidity of solvable groups
| dc.creator | Eskin, Alex | |
| dc.creator | Fisher, David | |
| dc.creator | Whyte, Kevin | |
| dc.date | 2005-11-27 | |
| dc.date | 2006-07-07 | |
| dc.date.accessioned | 2026-07-07T06:51:46Z | |
| dc.date.available | 2026-07-07T06:51:46Z | |
| dc.description | In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasi-isometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to $R \ltimes R^n$ where the semidirect product is defined by a diagonalizable matrix of determinant one with no eigenvalues on the unit circle. Our approach to these problems is to first classify all self quasi-isometries of the solvable Lie group. Our classification of self quasi-isometries for $R \ltimes \R^n$ proves a conjecture made by Farb and Mosher in [FM4]. Our techniques for studying quasi-isometries extend to some other classes of groups and spaces. In particular, we characterize groups quasi-isometric to any lamplighter group, answering a question of de la Harpe [dlH]. Also, we prove that certain Diestel-Leader graphs are not quasi-isometric to any finitely generated group, verifying a conjecture of Diestel and Leader from [DL] and answering a question of Woess from [SW],[Wo1]. We also prove that certain non-unimodular, non-hyperbolic solvable Lie groups are not quasi-isometric to finitely generated groups. The results in this paper are contributions to Gromov's program for classifying finitely generated groups up to quasi-isometry [Gr2]. We introduce a new technique for studying quasi-isometries, which we refer to as "coarse differentiation". | |
| dc.description | 19 pages, 5 figures. One theorem added concerning quasi-isometric rigidity of three manifold groups | |
| dc.identifier | https://arxiv.org/abs/math/0511647 | |
| dc.identifier | http://arxiv.org/abs/math/0511647 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/105096 | |
| dc.subject | Group Theory | |
| dc.subject | Geometric Topology | |
| dc.subject | Metric Geometry | |
| dc.title | Quasi-isometries and rigidity of solvable groups | |
| dc.type | text |