Centralisers and the virtually cyclic dimension of $\mathrm{Out}(F_N)$

Abstract

We prove that the virtually cyclic (geometric) dimension of the finite index congruence subgroup ${\mathrm{IA}}_N(3)$ of $\mathrm{Out}(F_N)$ is $2N-2$. From this we deduce the virtually cyclic dimension of $\mathrm{Out}(F_N)$ is finite. Along the way we prove Lück's property (C) holds for $\mathrm{Out}(F_N)$, we prove that the commensurator of a cyclic subgroup of ${\mathrm{IA}}_N(3)$ equals its centraliser, we give an ${\mathrm{IA}}_N(3)$ analogue of various exact sequences arising from reduction systems for mapping class groups, and give a near complete description of centralisers of infinite order elements in ${\mathrm{IA}}_3(3)$.

35 pages, comments welcome