Relative outer automorphisms of free groups

Erika Meucci

paper

Relative outer automorphisms of free groups

arXiv:1010.4753

Abstract

Let $A_{1}, ..., A_{k}$ be a system of free factors of $F_{n}$ . The group of relative automorphisms $A u t (F_{n}; A_{1}, ..., A_{k})$ is the group given by the automorphisms of $F_{n}$ that restricted to each $A_{i}$ are conjugations by elements in $F_{n}$ . The group of relative outer automorphisms is defined as $O u t (F_{n}; A_{1}, ..., A_{k}) = A u t (F_{n}; A_{1}, ..., A_{k}) / I nn (F_{n})$ , where $I nn (F_{n})$ is the normal subgroup of $A u t (F_{n})$ given by all the inner automorphisms. We define a contractible space on which $O u t (F_{n}; A_{1}, ..., A_{k})$ acts with finite stabilizers and we compute the virtual cohomological dimension of this group.

24 pages, 16 figures, corrected typos, revised argument in section 5, results unchanged