Hidden torsion, 3‐manifolds, and homology cobordism. (24th February 2013) - Cha, Jae Choon; Orr, Kent E.
Abstract : This paper continues our exploration of homology cobordism of 3‐manifolds using our recent results on Cheeger–Gromov ρ‐invariants associated to amenable representations. We introduce a new type of torsion in 3‐manifold groups that 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.