Hidden torsion, 3-manifolds, and homology cobordism

TL;DR

This paper introduces hidden torsion in 3-manifold groups, constructs hyperbolic 3-manifolds with this property, and uses Cheeger-Gromov invariants to distinguish their homology cobordism classes, answering a longstanding question.

Contribution

It defines hidden torsion and local hidden torsion, constructs infinitely many hyperbolic 3-manifolds with these properties, and applies Cheeger-Gromov invariants to distinguish their homology cobordism classes.

Findings

Constructed infinitely many hyperbolic 3-manifolds with local hidden torsion.

Showed these manifolds are not homology cobordant despite previous invariants failing to distinguish them.

Provided an answer to a question about the transfinite lower central series of homology cobordant 3-manifold groups.

Abstract

This paper continues our exploration of homology cobordism of 3-manifolds using our recent results on Cheeger-Gromov rho-invariants associated to amenable representations. We introduce a new type of torsion in 3-manifold groups we call hidden torsion, and an algebraic approximation we call local hidden torsion. We construct infinitely many hyperbolic 3-manifolds which have local hidden torsion in the transfinite lower central subgroup. By realizing Cheeger-Gromov invariants over amenable groups, we show that our hyperbolic 3-manifolds are not pairwise homology cobordant, yet remain indistinguishable by any prior known homology cobordism invariants. Additionally we give an answer to a question about transfinite lower central series of homology cobordant 3-manifold groups, asked by T. D. Cochran and M. H. Freedman.