Geometry of the smallest 1-form Laplacian eigenvalue on hyperbolic manifolds

TL;DR

This paper explores the relationship between small eigenvalues of the 1-form Laplacian and the geometric complexity of cycles in hyperbolic manifolds, with implications for homology growth in 3-manifold families.

Contribution

It establishes a link between Laplacian eigenvalues and cycle complexity, proposing applications to homology growth in hyperbolic 3-manifolds.

Findings

Small eigenvalues correspond to certain geodesic properties.

Potential applications to homology growth in hyperbolic 3-manifolds.

Provides a new perspective on the geometry of hyperbolic manifolds.

Abstract

We relate small 1-form Laplacian eigenvalues to relative cycle complexity on closed hyperbolic manifolds: small eigenvalues correspond to closed geodesics no multiple of which bounds a surface of small genus. We describe potential applications of this equivalence principle toward proving optimal torsion homology growth in families of hyperbolic 3-manifolds Benjamini-Schramm converging to $\mathbb{H}^3.$