Homogeneity groups of ends of open $3$-manifolds

Dennis J. Garity and Dušan Repovš

Vol. 269 (2014), No. 1, 99–112

DOI: 10.2140/pjm.2014.269.99

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 whose homogeneity group is 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 the local genus of points in Cantor sets and about the geometric index are also used.

Keywords

open 3-manifold, rigidity, manifold end, geometric index, Cantor set, homogeneity group, abelian group, defining sequence

Mathematical Subject Classification 2010

Primary: 54E45, 57M30, 57N12

Secondary: 57N10, 54F65

Milestones

Received: 11 September 2012

Revised: 19 July 2013

Accepted: 23 July 2013

Published: 15 July 2014