This paper introduces a new topological space associated with a nonabelian free group F n of rank n and a malnormal subgroup system A of F n , called the space of currents relative to 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 A are dense in the space of currents relative to A. 1

This paper, which is the second of a series of three papers, studies dynamical properties of elements of OutpF n q, the outer automorphism group of a nonabelian free group F n . We prove that, for every exponentially growing outer automorphism of F n , there exists a preferred compact topological space, the space of currents relative to a malnomal subgroup system, on which φ acts by homeomorphism with a North-South dynamics behavior. 1

This paper studies the geometric rigidity of the universal Coxeter group of rank n, which is the free product W n of n copies of Z{2Z. We prove that for n ě 4 the group of symmetries of the spine of the Guirardel-Levitt outer space of W n is reduced to the outer automorphism group OutpW n q. 1