Failure of the well-rounded retract for outer space and Teichmüller space

Abstract

The well-rounded retract for S L n ( Z ) SL_n(\mathbb {Z}) is defined as the set of flat tori of unit volume and dimension n 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 X_n of metric graphs of rank n n and the Teichmüller space T g \mathcal {T}_g of closed hyperbolic surfaces of genus g 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 T g \mathcal {T}_g by Thurston) are spines for X n X_n but that their dimension is larger than the virtual cohomological dimension of O u t ( F n ) Out(F_n) in general.