Twisted conjugacy classes in Symplectic groups, Mapping class groups and Braid groups(including an Appendix written with Francois Dahmani)

Alexander Fel'shtyn; Daciberg L. Gonçalves

paper

Twisted conjugacy classes in Symplectic groups, Mapping class groups and Braid groups(including an Appendix written with Francois Dahmani)

arXiv:0708.2628

Abstract

We prove that the symplectic group $S p (2 n, Z)$ and the mapping class group $M o d_{S}$ of a compact surface $S$ satisfy the $R_{\infty}$ property. We also show that $B_{n} (S)$ , the full braid group on $n$ -strings of a surface $S$ , satisfies the $R_{\infty}$ property in the cases where $S$ is either the compact disk $D$ , or the sphere $S^{2}$ . 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.

21 pages, with Appendix