We present a construction which associates to differential equations discrete groups. In order to establish this relation we use automata and random walks on ultra discrete limits. We discuss related results concerning von Neumann dimension and L2 Betti numbers of closed manifolds. Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006) The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page