Free groups as end homogeneity groups of $3$-manifolds

TL;DR

This paper constructs specific open 3-manifolds whose end symmetry groups are any given finitely generated free group, extending previous results to a broader class of groups.

Contribution

It introduces a method to realize any finitely generated free group as the end homogeneity group of an irreducible open 3-manifold, broadening the scope of known end symmetry groups.

Findings

Constructed manifolds with end homogeneity groups isomorphic to any finitely generated free group.

Extended previous results from finitely generated abelian groups to free groups.

Provided a method linking Cayley graph properties to manifold end homogeneity groups.

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 $G$.