For hyperbolic manifolds, we study two quantifications of being a homology 3-sphere, one geometric and the other topological: the spectral gap for the Laplacian on coclosed 1-forms and the size of the first torsion homology group.
We first produce different examples of sequences of hyperbolic homology 3-spheres with volume going to infinity and with a uniform spectral gap on coclosed 1-forms.
This answers a question of Lin-Lipnowski which they asked as a step towards constructing infinitely many examples of hyperbolic 3-manifolds that do not admit any irreducible solutions to the Seiberg-Witten equations.
We then focus on the relation between a sequence having a uniform spectral gap, and exponential growth of torsion homology in that sequence. For arithmetic towers the work of Bergeron-Sengun-Venkatesh conjecturally suggests a precise such relation.
We show that for any sequence of closed hyperbolic rational homology 3-spheres that converges to a tame manifold with at least one end, if the sequence has a uniform spectral gap for coexact 1-forms, then the torsion homology grows exponentially.
This is based on joint work with Anshul Adve, Vikram Giri, Ben Lowe and Jonathan Zung.
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Ahmed Abbes
