#### Volume 14, issue 3 (2014)

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

### Matthew C B Zaremsky

Algebraic & Geometric Topology 14 (2014) 1845–1879
##### 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 $m>\left(3i-1\right)∕2$.

##### Keywords
partially symmetric automorphism, homological stability
##### Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 20F28, 57M07