  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.
