Counting paths in solvable groups and quasi-isometry rigidity

I will begin by discussing joint work with Eskin and Whyte on quasi-isometric rigidity of certain solvable groups. This work lead us to conjecture quasi-isometric rigidity of all polycyclic groups. I will give some indication of difficulties and progress on that problem. The main new results, joint with Irine Peng, involve describing the structure of certain graphs arising in analysis on these groups and showing that they are fairly homogeneous.