Twisted conjugacy classes in symplectic groups, mapping class groups and braid groups (with an appendix written jointly with Francois Dahmani)

We prove that the symplectic group Sp(2n, Z) and the mapping class group Mod(S) of a compact surface S satisfy the R(infinity) property. We also show that B(n)(S), the full braid group on n-strings of a surface S, satisfies the R(infinity) property in the cases where S is either the compact disk D, or the sphere S(2). This means that for any automorphism phi of G, where G is one of the above groups, the number of twisted phi-conjugacy classes is infinite.

Braid groups of non-orientable surfaces and the Fadell-Neuwirth short exact sequence

Let M be a compact, connected non-orientable surface without boundary and of genus g >= 3. We investigate the pure braid groups P,(M) of M, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence 1 -> P(m)(M \ {x(1), ..., x(n)}) hooked right arrow P(n+m)(M) (P*) under right arrow P(n)(M) -> 1, where m, n >= 1, and p* is the homomorphism which corresponds geometrically to forgetting the last m strings. This problem is equivalent to that of the existence of a section for the associated fibration p: F(n+m)(M) -> F(n)(M) of configuration spaces, defined by p((x(1), ..., x(n), x(n+1), ..., x(n+m))) = (x(1), ..., x(n)). We show that p and p* admit a section if and only if n = 1. Together with previous results, this completes the resolution of the splitting problem for surface pure braid groups. (C) 2009 Elsevier B.V. All rights reserved.