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Let F_n be the free group of rank n, with generating set S=\{x_1,...,x_n\}.An automorphism \phi of F_n is called symmetric if for each 1\leq i\leq n,\phi(x_i) is conjugate to x_j or x_j^{-1} for some 1\leq j\leq n. Let \SigmaAut(F_n) be the group of symmetric automorphisms. We prove that the inclusion\Sigma Aut(F_n) \rightarrow \Sigma Aut(F_{n+1}) induces an isomorphism inrational homology for n>(3i-1)/2.

Rational homological stability for groups of symmetric automorphisms of free groups