Hyperbolic 3-manifolds with uniform spectral gap for coclosed 1-forms

TL;DR

This paper constructs sequences of hyperbolic 3-manifolds with large volume and uniform spectral gaps for coclosed 1-forms, revealing new relationships between spectral properties and torsion homology growth.

Contribution

It provides the first examples of hyperbolic homology spheres with uniform spectral gaps, answering a previously open question, and explores the connection between spectral gaps and torsion homology growth.

Findings

Constructed sequences with uniform spectral gap and increasing volume.

Proved that torsion homology growth must be unbounded in such sequences.

Showed exponential torsion growth in manifolds with bounded rank and spectral gap.

Abstract

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.