Twisted Conjugacy Classes in Symplectic Groups, Mapping Class Groups and Braid Groups (with an Appendix Written Jointly with Francois Dahmani)

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We prove that the symplectic group Sp(2n, ℤ) and the mapping class group ModS of a compact surface S satisfy the R property. We also show that Bn(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 S2. This means that for any automorphism φ of G, where G is one of the above groups, the number of twisted φ-conjugacy classes is infinite.