Failure of the well-rounded retract for Outer space and Teichm\"uller space

Abstract

The well-rounded retract for $\mathrm{SL}_n(\mathbb{Z})$ is defined as the set of flat tori of unit volume and dimension $n$ whose systoles generate a finite-index subgroup in homology. This set forms an equivariant spine of minimal dimension for the space of flat tori. For both the Outer space $X_n$ of metric graphs of rank $n$ and the Teichm\"uller space $\mathcal{T}_g$ of closed hyperbolic surfaces of genus $g$, we show that the literal analogue of the well-rounded retract does not contain an equivariant spine. We also prove that the sets of graphs whose systoles fill either topologically or geometrically (two analogues of a set proposed as a spine for $\mathcal{T}_g$ by Thurston) are spines for $X_n$ but that their dimension is larger than the virtual cohomological dimension of $\mathrm{Out}(F_n)$ in general.