Mathematics Colloquium, Fall 2007

Coarse differentiation of quasi-isometries and rigidity for solvable groups

In the early 80's Gromov initiated a program to study finitely generated groups up to quasi-isometry. This program was motivated by rigidity properties of lattices in Lie groups. A lattice Gamma in a group G is a discrete subgroup where the quotient G/Gamma has finite volume. Gromov's own major theorem in this direction is a rigidity result for lattices in nilpotent Lie groups. In the 1990's, a series of dramatic results led to the completion of the Gromov program for lattices in semisimple Lie groups. The next natural class of examples to consider are lattices in solvable Lie groups, and even results for the simplest examples were elusive for a considerable time. I will discuss joint work with Eskin and Whyte in which we prove the first results on quasi-isometry classification of lattices in solvable Lie groups. The results are proven by a method of coarse differentiation, which I will outline. I will also describe some interesting results concerning groups quasi-isometric to homogeneous graphs that follow from the same methods.