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