A LOWER ESTIMATE FOR CENTRAL PROBABILITIES ON POLYCYCLIC GROUPS

We give a lower estimate for the central value mu*n(e) of the nth convolution power mu*...*mu of a symmetric probability measure mu on a polycyclic group G of exponential growth whose support is finite and generates G. We also give a similar large time diagonal estimate for the fundamendal solution of the equation (partial derivative/partial derivative t + L)u = 0, where L is a left invariant sub-Laplacian on a unimodular amenable Lie group G of exponential growth.