Centralisers of linear growth automorphisms of free groups

Abstract

In this note we investigate the centraliser of a linearly growing element of $\mathrm{Out}(F_n)$ (that is, a root of a Dehn twist automorphism), and show that it has a finite index subgroup mapping onto a direct product of certain "equivariant McCool groups" with kernel a finitely generated free abelian group. In particular, this allows us to show it is VF and hence finitely presented.

23 pages, 2 figures; v2: final version, to appear in Mathematical Proceedings of the Cambridge Philosophical Society; new introduction and some reordering