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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 Fn+m be the free group of rank n + m, with generators x1,,xn+m. An automorphism ϕ of Fn+m is called partially symmetric if for each 1 i m, ϕ(xi) is conjugate to xj or xj1 for some 1 j m. Let ΣAutnm be the group of partially symmetric automorphisms. We prove that for any m 0 the inclusion ΣAutnm ΣAutn+1m induces an isomorphism in rational homology for dimensions i satisfying n (3(i + 1) + m)2, with a similar statement for the groups PΣAutnm of pure partially symmetric automorphisms. We also prove that for any n 0 the inclusion ΣAutnm ΣAutnm+1 induces an isomorphism in rational homology for dimensions i satisfying m > (3i 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/