Fixed points, splittings and division rings

Let G be a free group of rank N, let f be an automorphism of G and let Fix(f) be the corresponding subgroup of fixed points. Bestvina and Handel showed that the rank of Fix(f) is at most N, for which they developed the theory of train track maps on free groups. Different arguments were provided later on by Sela, Paulin and Gaboriau-Levitt-Lustig. In this talk, we present a new proof which involves the Linnell division ring of G. We also discuss how our approach relates to previous ones and how it gives new insight into variations of the problem.