On the exponentiality of the connected isometry group of a Damek-Ricci space

We prove that the connected isometry group of a non symmetric (non compact) irreducible Damek-Ricci space has a surjective exponential map if and only if the center of the associated Heisenberg type algebra has dimension less than or equal to 5. This result is analogous to (and extends) the results proved by the second author concerning the exponential map of the connected isometry group of an irreducible, rank one, classical, symmetric space of non compact type and that of the authors of [9] in the case of the Cayley plane to all irreducible non compact DR spaces.