Rational homological stability for groups of partially symmetric automorphisms of free groups

Details Rational homological stability for groups of partially symmetric automorphisms of free groups Rational homological stability for groups of partially symmetric automorphisms of free groups

2012-03-21

arxiv.org/abs/1203.4845v1

Mathematics

Group Theory

17 pages, 2 figures

Scientific paper

Let F_{n+m} be the free group of rank n+m, with generating set S=\{x_1,...,x_{n+m}\}. An automorphism \phi of F_{n+m}$ is called partially symmetric if for each 1\leq i\leq m, \phi(x_i) is conjugate to x_j or x_j^{-1} for some 1\leq j\leq m. Let \Sigma Aut_n^m$ be the group of partially symmetric automorphisms. Using the action of this group on a certain subcomplex of the spine of Auter space, we prove that for any m\geq 0 the inclusion \Sigma Aut_n^m \rightarrow \Sigma Aut_{n+1}^m induces an isomorphism in rational homology for n\geq (3(i+1)+m)/2, and for any n\geq 0 the inclusion \Sigma Aut_n^m \to \Sigma Aut_n^{m+1} induces an isomorphism in rational homology for m>(3(i+1)-n)/2. We also show that for any m\geq 0, the stable rational homology in n is trivial.

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