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

Zaremsky, Matthew C B¹

¹Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902, USA

Algebraic and Geometric Topology, Tome 14 (2014) no. 3, pp. 1845-1879

Résumé

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 xj−1 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.

Détail

DOI : 10.2140/agt.2014.14.1845

Classification : 20F65, 20F28, 57M07
Keywords: partially symmetric automorphism, homological stability

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