Separable homology of graphs and the separability complex

Abstract

We introduce the separability complex, a one-complex associated to a finite regular cover of the rose and show that it is connected if and only if the fundamental group of the associated cover is generated by its intersection with the set of elements in proper free factors of $\mathbf{F}_n$. The separability complex admits an action of $\mathrm{Out}(\mathbf{F}_n)$ by isometries if the associated cover corresponds to a characteristic subgroup of $\mathbf{F}_n$. We prove that the separability complex of the rose has infinite diameter and is nonhyperbolic, implying it is not quasi-isometric to the free splitting complex or the free factor complex. As a consequence, we obtain that the Cayley graph of $\mathbf{F}_n$ with generating set consisting of all primitive elements of $\mathbf{F}_n$ is nonhyperbolic.