Free groups as end homogeneity groups of 3-manifolds

For every finitely generated free group F , we construct an irreducible open 3-manifold MF whose end set is homeomorphic to a Cantor set, and with the end homogeneity group of MF isomorphic to F . The end homogeneity group is the group of all selfhomeomorphisms of the end set that extend to homeomorphisms of the entire 3-manifold. This extends an earlier result that constructs, for each finitely generated abelian group G, an irreducible open 3-manifold MG with end homogeneity group G. The method used in the proof of our main result also shows that if G is a group with a Cayley graph in R such that the graph automorphisms have certain nice extension properties, then there is an irreducible open 3-manifold MG with end homogeneity group G.