  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.
