Tianyi Zheng

In the first part of this dissertation we determine the behavior of the return probability of simple random walks on the free solvable group Sd,r of derived length d on r generators and some other related groups. In the second part, we study the decay of convolution powers of a large family µS,a of measures on finitely generated nilpotent groups. Here, S = (s1, . . . , sk) is a generating k-tuple of group elements and a = (α1, . . . , αk) is a k-tuple of reals in the interval (0, 2). The symmetric probability measure µS,a is supported by S∗ = {smi, 1 ≤ i ≤ k, m ∈ Z} and gives probability proportional to

(1 + m)−αi−1

to s±m, i = 1, . . . , k, m ∈ N. We determine the behavior of the probability of return µ(n) S,a(e) as n tends to infinity. This behavior depends in somewhat subtle ways on interactions between the k-tuple a and the positions of the generators si within the lower central series Gj = [Gj−1, G], G1 = G. In the third part, we prove tightness properties of some random walks on groups of polynomial volume growth
driven by spread-out measures, including the measures µS,a studied in the second part.