Centralisers of polynomially growing automorphisms of free groups

Abstract

The main theorem of this thesis asserts that many centralisers in the automorphism groups Aut(F_n) and Out(F_n) of the free group F_n satisfy finiteness property VF, i.e. these centralisers have a finite index subgroup with a finite polyhedron as a classifying space. I first introduce higher graphs of groups and their automorphisms, which generalise the well-known graphs of groups. Important automorphisms are higher Dehn twists. I then use the structure of the automorphism group of the higher graph of groups to show that centralisers of higher Dehn twist automorphisms in Out(F_n) and Aut(F_n) are of type VF. This includes all polynomially growing automorphisms up to passing to powers. Finally, I compute explicit abelianisations of some centralisers, which give information on translation lengths in isometric CAT(0) actions.