I will explain a few problems concerning distributions of endpoints of certain types of paths in nilpotent and solvable groups.

One can think of this as looking at the distribution of a random walk at large enough finite times. I will focus on a few special cases of quite general results, just to avoid too much notation and terminology. I will also explain why these problems arise naturally in the study of quasi-isometries of solvable Lie groups. This is joint work with Irine Peng.