Geometric and Functional Analysis

, Volume 19, Issue 5, pp 1426-1467

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Intersection Form, Laminations and Currents on Free Groups

  • Ilya KapovichAffiliated withDepartment of Mathematics, University of Illinois at Urbana-Champaign Email author 
  • , Martin LustigAffiliated withMathématiques (LATP), Université Paul Cézanne -Aix Marseille III


Let F be a free group of rank N ≥ 2, let μ be a geodesic current on F and let T be an \({\mathbb{R}}\)-tree with a very small isometric action of F. We prove that the geometric intersection number \({{\langle T,\mu \rangle}}\) is equal to zero if and only if the support of μ is contained in the dual algebraic lamination L 2(T) of T. Applying this result, we obtain a generalization of a theorem of Francaviglia regarding length spectrum compactness for currents with full support. We use the main result to obtain “unique ergodicity” type properties for the attracting and repelling fixed points of atoroidal iwip elements of Out(F) when acting both on the compactified outer Space and on the projectivized space of currents. We also show that the sum of the translation length functions of any two “sufficiently transverse” very small F-trees is bilipschitz equivalent to the translation length function of an interior point of the outer space. As another application, we define the notion of a filling element in F and prove that filling elements are “nearly generic” in F. We also apply our results to the notion of bounded translation equivalence in free groups.

Keywords and phrases

Free groups Outer space geodesic currents

2000 Mathematics Subject Classification

Primary 20F Secondary 57M 37B 37D