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

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We compute the Bieri-Neumann-Strebel invariants Σ 1 for the full and pure braid groups of the sphere S 2 , the real projective plane R P 2 , 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 Σ 1 ( P n ( M ) ) c is invariant. Furthermore, Σ 1 ( P n ( T ) ) c , and Σ 1 ( P n ( S 2 ) ) c (with n ≥ 5 ) are finite unions of circles, and Σ 1 ( P n ( K ) ) c is finite. This implies the existence of H⏧ Aut ( P n ( K ) ) with | Aut ( P n ( K ) ) : H | < ∞ such that R ( φ ) = ∞ for every φ ∈ H .