We prove that for k > 5 there does not exist a continuous map ∂CV(Fk) → PCurr(Fk) that is either Out(F k)-equivariant or Out(Fk)-anti-equivariant. Here ∂CV(Fk) is the 'length function' boundary of Culler-Vogtmann's Outer space CV(Fk), and PCurr(Fk) is the space of projectivized geodesic currents for Fk. We also prove that, if k ≥ 3, for the action of Out(Fk) on ℙCwrr(Fk) and for the diagonal action of Out(Fk) on the product space ∂CV(F k) × PCurr(Fk), there exist unique non-empty minimal closed Out(Fk)-invariant sets. Our results imply that for k ≥ 3 any continuous Out(Fk)-equivariant embedding of CV(Fk) into ℙCurr(Fk) (such as the Patterson-Sullivan embedding) produces a new compactification of Outer space, different from the usual 'length function' compactification CV(Fk) = CV(Fk) ∪ ∂CV(Fk).
ASJC Scopus subject areas
- Applied Mathematics