Topologically unrealizable automorphisms of free groups

Abstract:

Let $\phi :F \to F$ be an automorphism of a finitely generated free group. It has been conjectured (I heard it from Peter Scott) that the fixed subgroup of $\phi$ is always finitely generated. This is known to be so if $\phi$ has finite order [1], or if $\phi$ is realizable by a homeomorphism of a compact $2$-manifold with boundary [2]. Here we give examples of automorphisms $\phi$, no power of which is topologically realizable on any $2$-manifold; perhaps the simplest is the automorphism of the free group of rank 3, given by $\phi (x) = y$, $\phi (y) = z$, $\phi (z) = xy$.