I will report on work with Karen Vogtmann on the rational Euler characteristic of Out(F_n) and the moduli space of graphs. A similar study has been performed in the seminal 1986 work of Harer and Zagier on the Euler characteristic of the mapping class group and the moduli space of curves. I will review a topological field theory proof, due to Kontsevich, of Harer and Zagier´s result and illustrate how an analogous `renormalized` topological field theory argument can be applied to Out(F_n). Eventually, I will remark how an elaborate study of the entralizers of finite order elements of OutFn could lead to additional insights into the `naive' integral Euler characteristic of OutFn.
The Euler characteristic of Out(Fn) and renormalized topological field theory