Currents relative to a malnormal subgroup system

Abstract

This paper introduces a new topological space associated with a nonabelian free group $F_n$ of rank $n$ and a malnormal subgroup system $\mathcal{A}$ of $F_n$, called the space of currents relative to $\mathcal{A}$, which are $F_n$-invariant measures on an appropriate subspace of the double boundary of $F_n$. The extension from free factor systems as considered by Gupta to malnormal subgroup systems is necessary in order to fully study the growth under iteration of outer automorphisms of $F_n$, and requires the introduction of new techniques on cylinders. We in particular prove that currents associated with elements of $F_n$ which are not contained in a conjugate of a subgroup of $\mathcal{A}$ are dense in the space of currents relative to $\mathcal{A}$.