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 L\"uck'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 $\mathrm{ IA}_3(3)$.
Categories: math.GR math.AT math.GT