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

TL;DR

This paper establishes the finite virtually cyclic dimension of $ ext{Out}(F_N)$ and its subgroup $ ext{IA}_N(3)$, providing new insights into their algebraic structure and properties related to centralisers and reduction systems.

Contribution

It proves the virtually cyclic dimension of $ ext{IA}_N(3)$ is $2N-2$, shows $ ext{Out}(F_N)$ has finite virtually cyclic dimension, and extends several structural results from mapping class groups.

Findings

Virtually cyclic dimension of $ ext{IA}_N(3)$ is $2N-2$

Virtually cyclic dimension of $ ext{Out}(F_N)$ is finite

Lück's property (C) holds for $ ext{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 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 $\mathrm{IA}_3(3)$.