THE ACTIONS OF Out(Fk) ON THE BOUNDARY OF OUTER SPACE AND ON THE SPACE OF CURRENTS: MINIMAL SETS AND EQUIVARIANT INCOMPATIBILITY
نویسنده
چکیده
We prove that for k ≥ 5 there does not exist a continuous map ∂CV (Fk) → PCurr(Fk) that is either Out(Fk)-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 PCurr(Fk) and for the diagonal action of Out(Fk) on the product space ∂CV (Fk)× 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).
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