Fixed subgroups of automorphisms of free by finite groups: an extension of Cooper's proof (Q1092162)

The author generalizes the known result (first proved by Gersten) that the fixed subgroup of an automorphism of a finitely generated free group is finitely generated to finite extensions of free groups. That the fixed subgroup of an automorphism of a finite extension of a finitely generated free group is finitely generated comes as a corollary of the above mentioned result of Gersten. The author's result is more general. In fact he proves that if G is a finite extension of a free group (of arbitrary rank), \(f\in Aut(G)\) and \(G_ 1\) a finitely generated subgroup of G then \(Fix(f)\cap G_ 1\) is finitely generated. The proof makes use of the fact that the group G acts on a tree so that the stabilizers of vertices are finite groups of bounded order and is analogous to D. Cooper's method for the free group. The paper contains also a proof, along the same lines, of the fact that a finite extension of a free group has the Howson property.