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.

70 pages. Updated for release of Parts III and IV. All other parts including the research announcement are found on this arXiv