Séminaire de Mathématique
# Hyperbolic 3-Manifolds with Spectral Gap for Coclosed 1-Forms and Torsion Homology Growth

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Amphithéâtre Léon Motchane (IHES)
### Amphithéâtre Léon Motchane

#### IHES

Le Bois Marie
35, route de Chartres
CS 40001
91893 Bures-sur-Yvette Cedex

Description

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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Organized by

Ahmed Abbes

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