On the \(p\)-primary cohomology of \(\text{Out}(F_n)\) in the \(p\)-rank one case
type: Item
instance of: scholarly article
title: On the \(p\)-primary cohomology of \(\text{Out}(F_n)\) in the \(p\)-rank one case
zbMATH Open document ID: 0986.20050
DOI: 10.1016/S0022-4049(99)00071-7
published in: Journal of Pure and Applied Algebra
publication date: 2001-03-12T00:00:00Z
review text: Let \(p\) be an odd prime, \(F_n\) a free group of rank \(n\) and \(\text{Out}(F_n)=\Aut(F_n)/\text{Inn}(F_n)\) its group of outer automorphisms. \textit{H. H. Glover}, \textit{G. Mislin} and \textit{S. N. Voon} [Lond. Math. Soc. Lect. Note Ser. 252, 161-169 (1998; Zbl 0907.55007)] proved that \(\text{Out}(F_n)\) has \(p\)-rank one if and only if \(p-1\leq n<2(p-1)\). In this paper, the authors compute the \(p\)-primary part of the cohomology of \(\text{Out}(F_n)\) for \(n=p-1\), \(p\) and \(p+1\) in the range above the virtual cohomological dimension of \(\text{Out}(F_n)\) (\(0\) if \(n=1\) and \(2n-3\) if \(n>1\)). The way of doing it is using a Leray spectral sequence of the form \(\overline\sigma\in{\mathcal O}(X_n)\prod H^*(\text{stab}(\sigma);\mathbb{Z}_{(p)})\Rightarrow H^*(\text{Out}(F_n);\mathbb{Z}_{(p)})\), where \(X_n\) is the simplicial spine of outer space, that is a \((2n-3)\)-dimensional contractible simplicial complex on which \(\text{Out}(F_n)\) acts with finite cell stabilizers, and \(\overline\sigma\) runs over the set of orbits \({\mathcal O}(X_n)\) of simplices \(\sigma\subset X_n\). The existence of admissible graphs with free \(\mathbb{Z}/p\)-symmetry makes the case \(n=p+1\) the most complicated of the three.
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 55N35
zbMATH DE Number: 1526262
zbMATH Keywords: free groups
zbMATH Keywords: outer automorphisms
zbMATH Keywords: cohomology
zbMATH Keywords: Leray spectral sequences
zbMATH Keywords: simplicial complexes
zbMATH Keywords: symmetric groups
zbMATH Keywords: dihedral groups
zbMATH Keywords: metacyclic groups
zbMATH Keywords: virtual cohomological dimension
author: Henry H. Glover
author: Guido Mislin
reviewed by: M. Ladra
MaRDI profile type: Publication
cites work: The integral cohomology of \(\text{Out}_ +(F_ 3)\)
cites work: Q3708977
cites work: Moduli of graphs and automorphisms of free groups
cites work: Q4400498
cites work: A generating function for the Euler characteristic of \(Out(F_ n)\)
full work available at URL: s0022-4049(99)00071-7
OpenAlex ID: W2075730884
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description: scientific article; zbMATH DE number 1526262