\(\mathbb{R}\)-trees and laminations for free groups. III: Currents and dual \(\mathbb{R}\)-tree metrics.

*(English)*Zbl 1200.20018Summary: 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).

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 |