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

Zaremsky, Matthew C B

doi:10.2140/agt.2014.14.1845

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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

DOI: 10.2140/agt.2014.14.1845

Abstract

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

Keywords

partially symmetric automorphism, homological stability

Mathematical Subject Classification 2010

Primary: 20F65

Secondary: 20F28, 57M07

References

Publication

Received: 17 January 2013

Revised: 21 October 2013

Accepted: 15 November 2013

Published: 29 May 2014

Authors

Matthew C B Zaremsky
Department of Mathematical Sciences Binghamton University Binghamton, NY 13902 USA
http://www.math.binghamton.edu/zaremsky/

Volume 14, issue 3 (2014)