Intersection form, laminations and currents on free groups

TL;DR

This paper explores the relationship between intersection numbers, laminations, and currents in free groups, providing new characterizations, generalizations of length spectrum compactness, and insights into filling elements and translation equivalence.

Contribution

It establishes a criterion for when the intersection number vanishes, generalizes length spectrum compactness results, and introduces the concept of filling elements in free groups.

Findings

Intersection number zero iff support of current is in dual lamination

Generalization of length spectrum compactness theorem

Filling elements are nearly generic in 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.