Hyperbolic 3-manifolds with uniform spectral gap for coclosed 1-forms
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.
Cite
@article{arxiv.2404.19039,
title = {Hyperbolic 3-manifolds with uniform spectral gap for coclosed 1-forms},
author = {Amina Abdurrahman and Anshul Adve and Vikram Giri and Ben Lowe and Jonathan Zung},
journal= {arXiv preprint arXiv:2404.19039},
year = {2024}
}