The BNS invariants of the braid groups and pure braid groups of some surfaces

We compute the Bieri-Neumann-Strebel invariants Sigma(1)for the full and pure braid groups of the sphere S-2, the real projective plane RP2, the torus T and the Klein bottle K. For M = T or M=K, n >= 2, we show that the action by homeomorphisms of Out(P-n(M)) on S(P-n(M)) contains certain permutations, under which Sigma(1)(P-n(M))(c) is invariant. Furthermore, Sigma(1)(P-n(T))c, and Sigma(1)(P-n(S-2))(c) (with n >= 5) are finite unions of circles, and Sigma(1)(P-n(K))(c) is finite. This implies the existence of H(sic)Aut(P-n(K)) with |Aut(P-n(K)):H|<infinity such that R(phi)=infinity for every phi is an element of H.