Abstract
We establish the lower bound p2t(e,e)scapexp(-t1/3), for the large times asymptotic behaviours of the probabilities p2t(e,e) of return to the origin at even times 2t, for random walks associated with finite symmetric generating sets of solvable groups of finite Prüfer rank. (A group has finite Prüfer rank if there is an integer r, such that any of its finitely generated subgroup admits a generating set of cardinality less or equal to r.)