Intersection form, laminations and currents on free groups

Abstract

Let $F_N$ be a free group of rank $N\ge 2$, let $\mu$ be a geodesic current on $F_N$ and let $T$ be an $\mathbb{R}$-tree with a very small isometric action of $F_N$. We prove that the geometric intersection number $<T,\mu >$ is equal to zero if and only if the support of $\mu$ 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. As another application, we define the notion of a \emph{filling} element in $F_N$ and prove that filling elements are "nearly generic" in $F_N$. We also apply our results to the notion of \emph{bounded translation equivalence} in free groups.

revised version, to appear in GAFA

References in corpus (3)

• $\mathbb{R}$-trees and laminations for free groups II: The dual lamination of an $\mathbb{R}$-tree
• Geometric Intersection Number and analogues of the Curve Complex for free groups
• $\mathbb{R}$-trees and laminations for free groups III: Currents and dual $\mathbb{R}$-tree metrics

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