Homogeneity Groups of Ends of 3-manifolds

TL;DR

This paper constructs specific 3-manifolds with Cantor set ends whose end homogeneity groups match any given finitely generated abelian group, using advanced topological techniques involving Antoine type Cantor sets.

Contribution

It introduces a method to realize any finitely generated abelian group as the end homogeneity group of a 3-manifold with Cantor set ends, expanding understanding of end structures.

Findings

Constructed 3-manifolds with prescribed end homogeneity groups

Developed techniques for embedding homogeneity groups in Antoine type Cantor sets

Extended Antoine Cantor set constructions using infinite chains

Abstract

For every finitely generated abelian group G, we construct an irreducible open 3-manifold $M_{G}$ whose end set is homeomorphic to a Cantor set and with end homogeneity group of $M_{G}$ isomorphic to G. The end homogeneity group is the group of self-homeomorphisms of the end set that extend to homeomorphisms of the 3-manifold. The techniques involve computing the embedding homogeneity groups of carefully constructed Antoine type Cantor sets made up of rigid pieces. In addition, a generalization of an Antoine Cantor set using infinite chains is needed to construct an example with integer homogeneity group. Results about local genus of points in Cantor sets and about geometric index are also used.