  We study two quantifications of being a homology sphere for hyperbolic
3-manifolds, one geometric and one topological: the spectral gap for the
Laplacian on coclosed 1-forms and the size of the first torsion homology group.
We first construct a sequence of closed hyperbolic integer homology spheres
with volume tending to infinity and a uniform coclosed 1-form spectral gap.
This answers a question asked by Lin--Lipnowski. We also find sequences of
hyperbolic rational homology spheres with the same properties that
geometrically converge to a tame limit manifold. Moreover, we show that any
such sequence must have unbounded torsion homology growth. Finally we show that
a sequence of closed hyperbolic rational homology 3-spheres with uniformly
bounded rank and a uniform coclosed 1-form spectral gap must have torsion
homology that grows exponentially in volume.
