TY - JOUR
T1 - The actions of Out(Fk) on the boundary of Outer space and on the space of currents
T2 - Minimal sets and equivariant incompatibility
AU - Kapovich, Ilya
AU - Lustig, Martin
PY - 2007/6/1
Y1 - 2007/6/1
N2 - 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).
AB - 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).
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U2 - 10.1017/S0143385706001015
DO - 10.1017/S0143385706001015
M3 - Article
AN - SCOPUS:34248654352
VL - 27
SP - 827
EP - 847
JO - Ergodic Theory and Dynamical Systems
JF - Ergodic Theory and Dynamical Systems
SN - 0143-3857
IS - 3
ER -