Automorphisms of free groups with boundaries

Abstract

The automorphisms of free groups with boundaries form a family of groups $A_{n,k}$ closely related to mapping class groups, with the standard automorphisms of free groups as $A_{n,0}$ and (essentially) the symmetric automorphisms of free groups as $A_{0,k}$. We construct a contractible space $L_{n,k}$ on which $A_{n,k}$ acts with finite stabilizers and finite quotient space and deduce a range for the virtual cohomological dimension of $A_{n,k}$. We also give a presentation of the groups and calculate their first homology group.