R-trees and laminations for free groups III: Currents and dual R-tree metrics

A geodesic lamination L on a closed hyperbolic surface S, when provided with a transverse measure μ, gives rise to a “dual R-tree” Tμ, together with an action of G = π1S on Tμ by isometries. A point of Tμ corresponds precisely to a leaf of the lift L̃ of L to the universal covering S̃ of S, or to a complementary component of L̃ in S̃. The G-action on T is induced by the G-action on S̃ as deck transformations. This construction is well known (see [Mor86]). It is also known [Sko96] that conversely, for every small isometric action of a surface group G = π1S on a minimal R-tree T there exists a “dual” measured lamination (L, μ) on S, i.e. one has T = Tμ up to a G-equivariant isometry. This beautiful correspondence has tempted geometers and group theorists to investigate possible generalizations, and the first one arises if one replaces the closed surface by a surface with boundary, and correspondingly the surface group G by a free group FN of finite rank N ≥ 2. A first glimpse of the potential problems can be obtained from two simultaneous but distinct identifications FN ∼= −→ π1S1 and FN ∼= −→ π1S2, thus obtaining actions of π1S1 on a tree T2 which are dual to a measured lamination on S2, but in general not dual to any measured lamination on the surface S1. Worse, using the index of an R-tree action by FN as introduced in [GL95], it is easily seen that for many (perhaps even “most”) small or very small Rtrees T with isometric FN -action there is no identification whatsoever of FN with the fundamental group of any surface that would make T dual to a lamination. An example of such trees are the forward limit trees Tα of