Bond thickenings of the simplicial boundary of Outer space

Abstract

We study the simplicial boundary $\partial \mathcal{F}\mathcal{S}$ of Culler-Vogtmann Outer space via thickenings defined by graph-theoretic connectivity. Let $C^{\prime }$ be the subcomplex of the free splitting complex obtained from $\partial \mathcal{F}\mathcal{S}$ by adding all stable graphs that are not $3$-edge connected, together with their faces. We prove that the inclusion $\partial \mathcal{F}\mathcal{S}\hookrightarrow C^{\prime }$ is $(2n-3)$-connected. The proof shows, more precisely, that adding graphs with cut vertices is a homotopy equivalence, while the only non-contractible fibres in the $2$-bond thickening occur over $\theta$-graphs. The result gives further evidence that $\partial \mathcal{F}\mathcal{S}$ may be $(2n-3)$-spherical, an $\mathrm{Out}(F_n)$-analogue of Rognes's connectivity conjecture for the common basis complex. It also gives a topological, universal-cover perspective that unifies several existing results about the commutative graph complex.

39 pages, 11 figures; v2: added a reference