Topologically unrealizable automorphisms of free groups

Source abstract

Let ϕ : F → F \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 ϕ \phi is always finitely generated. This is known to be so if ϕ \phi has finite order [ 1 ], or if ϕ \phi is realizable by a homeomorphism of a compact 2 2 -manifold with boundary [ 2 ]. Here we give examples of automorphisms ϕ \phi , no power of which is topologically realizable on any 2 2 -manifold; perhaps the simplest is the automorphism of the free group of rank 3, given by ϕ ( x ) = y \phi (x) = y , ϕ ( y ) = z \phi (y) = z , ϕ ( z ) = x y \phi (z) = xy .