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

Abstract

In this paper, we continue with the results in [12] and compute the group of quasiisometries for a subclass of split solvable unimodular Lie groups. Consequently, we show that any finitely generated group quasi-isometric to a member of the subclass has to be polycyclic and is virtually a lattice in an abelian-by-abelian solvable Lie group. We also give an example of a unimodular solvable Lie group that is not quasi-isometric to any finitely generated group, as well deduce some quasi-isometric rigidity results. 51F99; 22E40