Automorphisms of graphs, p-subgroups of \(Out(F_ n)\) and the Euler characteristic of \(Out(F_ n)\)
If \(F_ n\) is a free group of finite rank n, then each automorphism of \(F_ n\) induces in a natural way an automorphism on the free abelian group \(F_ n/F'_ n\cong Z\) n and therefore there exists a natural homomorphism of Aut \(F_ n\) onto GL(n,\({\mathbb{Z}})\). Since each inner automorphism is mapped onto the identity, this gives a homomorphism of Out \(F_ n\) onto GL(n,\({\mathbb{Z}})\). Here the authors, by analyzing isometry groups of graphs, study finite subgroups of \(Out(F_ n)\) and derive information about the powers of primes which divide the denominator of the rational Euler characteristic \(\chi (Out(F_ n))\) of \(Out(F_ n)\). In this way the authors get information on the kernel of the homomorphism from \(Out(F_ n)\) onto GL(n,\({\mathbb{Z}})\) for n even. For example it is proved that \(\chi (Out(F_ n))\) is non-zero whenever n is even. Details of the methods and other results are too technical to be included here.
scientific article; zbMATH DE number 4044088