Free groups as end homogeneity groups of 3-manifolds

Abstract

For every finitely generated free group $F$, we construct an irreducible open 3-manifold $M_F$ whose end set is homeomorphic to a Cantor set, and with the end homogeneity group of $M_F$ isomorphic to $F$ . The end homogeneity group is the group of all self-homeomorphisms 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 $M_G$ 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 ${\mathbb{R}}^3$ such that the graph automorphisms have certain nice extension properties, then there is an irreducible open 3-manifold $M_G$ with end homogeneity group $G$.