Poisson boundary of random walks and growth of groups

Nilpotent groups are the closest class of noncommutative groups to abelian groups. Many results on Euclidean spaces can be considered there. The celebrated Gromov polynomial growth theorem asserts that a finitely generated discrete group has polynomial growth if and only if it is virtually nilpotent. More generally, for compactly generated locally compact groups of polynomial growth, structure theorems are given in a series of papers by Losert. In this minicourse, we will explore random walk models on groups of polynomial growth, starting from simple random walks on discrete groups, to more general random walks on locally compact ones, walks of unbounded range, etc. We will explain techniques to prove various estimates, limit theorems, and some applications beyond polynomial growth.