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\(\mathbb{R}\)-trees and laminations for free groups. III: Currents and dual \(\mathbb{R}\)-tree metrics. (English) Zbl 1200.20018
Summary: We study the map which associates to a current its support (which is a lamination). We show that this map is \(\text{Out}(F_N)\)-equivariant, not injective, not surjective and not continuous. However it is semi-continuous and almost surjective in a suitable sense. Given an \(\mathbb{R}\)-tree \(T\) (with dense orbits) in the boundary of outer space and a current \(\mu\) carried by the dual lamination of \(T\), we define a dual pseudo-distance \(d_\mu\) on \(T\). When the tree and the current come from a measured geodesic lamination on a surface with boundary, the dual distance is the original distance of the tree \(T\). In general, such a good correspondence does not occur. We prove that when the tree \(T\) is the attractive fixed point of a non-geometric irreducible, with irreducible powers, outer automorphism, the dual lamination of \(T\) is uniquely ergodic and the dual distance \(d_\mu\) is either zero or infinite throughout \(T\).
For part II cf. the authors, ibid. 78, No. 3, 737-754 (2008; Zbl 1198.20023).

MSC:
20E05 Free nonabelian groups
20E08 Groups acting on trees
20F65 Geometric group theory
37B10 Symbolic dynamics
57M07 Topological methods in group theory
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