On the geometry of a proposed curve complex analogue for $Out(F_n)$

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 $\ESC$, equivalently known as the separating sphere complex. We characterize geodesic paths in its 1-skeleton 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 $\ESC$. This shows that $\ESC$: 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 $\ESC$ and other proposed curve complex analogues, including the regular free splitting complex $\FSC$, the (nontrivial intersection) free factorization complex $\FFZC$, and the free factor complex $\FFC$, leaving hope that some of these complexes are hyperbolic.