Subgroup decomposition in Out(F_n), Part II: A relative Kolchin theorem

Abstract

This is the second in a series of four papers (with research announcement posted on this arXiv) that together develop a decomposition theory for subgroups of Out(F_n). In this paper we relativize the "Kolchin-type theorem" from the work of Bestvina, Feighn, and Handel on the Tits alternative, which describes a decomposition theory for subgroups H of Out(F_n) all of whose elements have polynomial growth. The Relative Kolchin Theorem allows subgroups H whose elements have exponential growth, as long as all such exponential growth is cordoned off in some free factor system F which is invariant under every element of H. The conclusion is that a certain finite index subgroup of H has an invariant filtration by free factor systems going from F up to the full free factor system by individual steps each of which is a "one-edge extension". We also study the kernel of the action of Out(F_n) on homology with Z/3 coefficients, and we prove Theorem B from the research announcement, which describes strong finite permutation behavior of all elements of this kernel.