On the \(p\)-primary cohomology of \(\text{Out}(F_n)\) in the \(p\)-rank one case
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.
scientific article; zbMATH DE number 1526262
scientific article