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

Abstract

Let $F_{n+m}$ be the free group of rank $n+m$, with generators $x_1,\dots ,x_{n+m}$. An automorphism $\varphi$ of $F_{n+m}$ is called partially symmetric if for each $1\le i\le m$, $\varphi \left(x_i\right)$ is conjugate to $x_j$ or $x_j^{-1}$ for some $1\le j\le m$. Let $\Sigma {Aut}_n^m$ be the group of partially symmetric automorphisms. We prove that for any $m\ge 0$ the inclusion $\Sigma {Aut}_n^m\to \Sigma {Aut}_{n+1}^m$ induces an isomorphism in rational homology for dimensions $i$ satisfying $n\ge \left(3\left(i+1\right)+m\right)∕2$, with a similar statement for the groups $P\Sigma {Aut}_n^m$ of pure partially symmetric automorphisms. We also prove that for any $n\ge 0$ the inclusion $\Sigma {Aut}_n^m\to \Sigma {Aut}_n^{m+1}$ induces an isomorphism in rational homology for dimensions $i$ satisfying $m>\left(3i-1\right)∕2$.