## $$\mathbb{R}$$-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

Zbl 1198.20023
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