On the geometry of the edge splitting complex

Abstract

The group Out of outer automorphisms of the free group has been an object of active study for many years, yet its geometry is not well understood. Recently, effort has been focused on finding a hyperbolic complex on which Out acts, in analogy with the curve complex for the mapping class group. Here, we focus on one of these proposed analogues: the edge splitting complex $\mathcal{E}{\mathcal{S}}_n$, equivalently known as the separating sphere complex. We characterize geodesic paths in its 1-skeleton $\mathcal{E}{\mathcal{S}}_n^1$ algebraically, and use our characterization to find lower bounds on distances between points in this graph.

Our distance calculations allow us to find quasiflats of arbitrary dimension in $\mathcal{E}{\mathcal{S}}_n$. This shows that $\mathcal{E}{\mathcal{S}}_n$: is not hyperbolic, has infinite asymptotic dimension, and is such that every asymptotic cone is infinite dimensional. These quasiflats contain an unbounded orbit of a reducible element of Out. As a consequence, there is no coarsely Out-equivariant quasiisometry between $\mathcal{E}{\mathcal{S}}_n$ and other proposed curve complex analogues, including the regular free splitting complex $\mathcal{F}{\mathcal{S}}_n$, the (nontrivial intersection) free factorization complex ${\mathcal{F}\mathcal{F}}_n$, and the free factor complex ${\mathcal{F}}_n$.