  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".
