Stabilization for the Automorphisms of Free Groups with Boundaries

There is by now a significant literature on homological stability properties of various families of discrete groups, including matrix groups, mapping class groups of surfaces, and automorphism groups of free groups. The present paper is yet another contribution in this direction, for a family of groups An,k that can be thought of as relative versions of automorphism groups of free groups. These groups An,k arose first in [W] in studying the relation between mapping class groups of surfaces and automorphism groups of free groups. In the present paper we exploit the fact that An,k can be realized as a quotient of the mapping class group of a certain 3 manifold, and our proof of stability applies more generally to give a homological stability result for the analogous quotients of the mapping class groups of almost all compact 3 manifolds. The quickest definition of An,k is in terms of homotopy equivalences of graphs. Let Gn,k denote the graph shown in Figure 1 consisting of a wedge of n circles together with k distinguished circles joined by arcs to the basepoint.