Quasi-isometries and rigidity of solvable groups

Details Quasi-isometries and rigidity of solvable groups Quasi-isometries and rigidity of solvable groups

2005-11-27

arxiv.org/abs/math/0511647v3

Mathematics

Group Theory

19 pages, 5 figures. One theorem added concerning quasi-isometric rigidity of three manifold groups

Scientific paper

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".

Affiliated with

Also associated with

No associations

Rating

Quasi-isometries and rigidity of solvable groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Quasi-isometries and rigidity of solvable groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Quasi-isometries and rigidity of solvable groups will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-613175