Currents relative to a malnormal subgroup system (Q6632024)

Let \(F_{n}\) be a free nonabelian group of rank \(n\). This paper is the first one in a sequence of papers by the author, where he studies the exponential growth of elements of \(\mathrm{Out}(F_{n})\).\N\NIn particular, this paper introduces a new topological space associated with \(F_{n}\) 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 \textit{R. Gupta} [Conform. Geom. Dyn. 21, 319--352 (2017; Zbl 1400.20017)] 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. The author proves that currents associated with elements of \(F_{n}\) that are not contained in a conjugate of a subgroup of \(\mathcal{A}\) are dense in the space of currents relative to \(\mathcal{A}\).