Automorphisms, fixed points and division rings

Given an automorphism of a space or a group, it is natural to study the structure of its fixed-point set. One concrete problem in this theme is the following. Let G be a finitely generated free group or the fundamental group of a closed surface, and let F be an automorphism of G. Can the fixed-point subgroup of F be infinitely generated? Can it have a rank bigger than the rank of G? It follows from results of Nielsen-Thurston and Bestvina-Handel that the previous questions have negative answer. We discuss a new, more algebraic proof of these results using a special division ring associated to the group ring of G; and discuss potential generalisations. This ring emanates from the algebraic reformulation of the theory of L^{^2}-invariants (originally introduced by Atiyah) due to Linnell, Lück and others.