Relative outer automorphisms of free groups

The study of automorphism groups of free groups is old, but the geometric approach to these groups is relatively new. Outer space was introduced in 1986 by Culler and Vogtmann as a tool for studying the group Out(Fn) of outer automorphisms of a finitely-generated free group. This work is focused on special subgroups of Out(Fn) called relative outer automorphisms groups. Let A1; : : : ;Ak be a system of free factors of Fn. The group of relative automorphisms Aut(Fn;A1; : : : ;Ak) is the group given by the automorphisms of Fn that restricted to each Ai are conjugations by elements in Fn. The group of relative outer automorphisms is denoted by Out(Fn;A1; : : : ;Ak) and de fined as Aut(Fn;A1; : : : ;Ak)=Inn(Fn), where Inn(Fn) is the normal subgroup of Aut(Fn) given by all the inner automorphisms. First, we defi ne the relative outer space on which a relative outer automorphism group of a free group acts properly discontinuously and we compute the virtual cohomological dimension of relative outer automorphism groups of a free group. Then we introduce another space, the modifi ed relative outer space, and we analyze its geometry and its dynamics. As a consequence, the Contracting Geodesics Theorem follows. This powerful theorem and an induction on the free factor system are the ingredients in the proof of the main application: every embedding of a lattice in Out(Fn) has fi nite image.