Hidden torsion, 3-manifolds, and homology cobordism

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.

24 pages; a new theorem answering a question of Cochran and Freedman (Kirby List 3.78) added; referee's comments incorporated; to appear in J. Topology

References in corpus (5)

• Knot Concordance and Higher-Order Blanchfield Duality
• Homology cylinders and sutured manifolds for homologically fibered knots
• $L^2$-signatures, homology localization, and amenable groups
• Abelian quotients of monoids of homology cylinders
• Twisted homology cobordism invariants of knots in aspherical manifolds

Cited by in corpus (3)

• Concordance of links with identical Alexander invariants
• Non-concordant links with homology cobordant zero framed surgery manifolds
• Right exact group completion as a transfinite invariant of the homology equivalence