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

Abstract

We study the map which associates to a current its support (which is a lamination). We show that this map is 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 R-tree T (with dense orbits) in the boundary of outer space and a current µ carried by the dual lamination of T , we define a dual pseudo-distance d µ 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 µ is either zero or infinite throughout T .

Geometric intersection number and analogues of the curve complex for free groups

“…This is due to the fact that in this paper ' acts on R-trees in CV.F n / from the left, while [12] and [33] in one considers the right-action (compare the discussion in Section 2).…”

“…(3) An alternative proof (relying on the main result of [27]) for Proposition 7.7 below is given by Proposition 5.6 of [12].…”

“…Remark 7.6 (1) We would like to alert the reader that the forward limit tree of ' , denoted in [12] and [33] by T ' , is the tree T (and not T C ). This is due to the fact that in this paper ' acts on R-trees in CV.F n / from the left, while [12] and [33] in one considers the right-action (compare the discussion in Section 2).…”

“…The space Curr.F N / turns out to be a natural companion of the Outer space and contains additional valuable information about the geometry and dynamics of free group automorphisms. Examples of such applications can be found in Bonahon [8], Coulbois, Hilion and Lustig [12], Francaviglia [14], Kaimanovich, Kapovich and Schupp [22], Kapovich [23;24;25], Kapovich and Lustig [29], Kapovich and Nagnibeda [30], Martin [34] and other sources.…”

Intersection Form, Laminations and Currents on Free Groups

“…The space Curr (F ) turns out to be a natural companion of the outer space and contains additional valuable information about the geometry and dynamics of free group automorphisms. Examples of such applications can be found in [Bo3], [CouHL3], [F], [Ka3,4,5], [KaLu1], [KaN], [KKS], [M] and other sources. Kapovich proved [Ka4] that for F there does not exist a natural symmetric analogue of Bonahon's intersection number between two geodesic currents.…”