Quasi-isometries and Rigidity of Solvable Groups

Abstract

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 quasiisometric to R 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 R n proves a conjecture made by Farb and Mosher in [FM3].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 quasiisometric 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.

Locations

• Pure and Applied Mathematics Quarterly - View - PDF
• arXiv (Cornell University) - View
• CiteSeer X (The Pennsylvania State University) - View - PDF