Random walks on unimodular p-adic groups

In a recent paper Pittet and Saloff-Coste established the lower bound p2n(e,e)⩾cexp(-Cn1/3), n=1,2,… for the large times asymptotic behaviours of the probabilities p2n(e,e) of return to the origin at even times 2n, for random walks associated with finite symmetric generating sets of solvable groups of finite Prüfer rank and asked if a similar lower bound is available in the case of the semi-direct product Sol(Qp)=Qp2⋉Qp*. In this paper, we give an answer to this problem.