Stable Isoperimetric Ratios and the Hodge Laplacian of Hyperbolic Manifolds

TL;DR

This paper establishes a relationship between the first eigenvalue of the Hodge Laplacian on hyperbolic 3-manifolds and an isoperimetric ratio involving geodesic length and stable commutator length, with implications for spectral gaps.

Contribution

It refines previous estimates by linking spectral properties of the Hodge Laplacian to geometric ratios, and constructs sequences of manifolds with exponentially vanishing spectral gaps.

Findings

Eigenvalue bounds depend polynomially on volume and injectivity radius.

Existence of hyperbolic 3-manifolds with spectral gap vanishing exponentially.

Refinement of Lipnowski and Stern's estimates.

Abstract

We show that for a closed hyperbolic 3-manifold the size of the first eigenvalue of the Hodge Laplacian acting on coexact 1-forms is comparable to an isoperimetric ratio relating geodesic length and stable commutator length with comparison constants that depend polynomially on the volume and on a lower bound on injectivity radius, refining estimates of Lipnowski and Stern. We use this estimate to show that there exist sequences of closed hyperbolic 3-manifolds with injectivity radius bounded below and volume going to infinity for which the 1-form Laplacian has spectral gap vanishing exponentially fast in the volume.