$L^2$-Cohomology of Geometrically Infinite Hyperbolic 3-Manifolds

TL;DR

This paper investigates the $L^2$-cohomology of topologically tame hyperbolic 3-manifolds, focusing on the existence of harmonic 1-forms and the spectral properties of the Laplacian.

Contribution

It provides new results on the presence of nonzero $L^2$ harmonic 1-forms and the spectral placement of zero in the Laplacian spectrum for these manifolds.

Findings

Nonzero $L^2$ harmonic 1-forms may or may not exist.

Zero can lie in the spectrum of the Laplacian on (1-forms)/Ker(d).

Results depend on the topological and geometric properties of the manifold.

Abstract

We give results on the following questions about a topologically tame hyperbolic 3-manifold M : 1. Does M have nonzero square-integrable harmonic 1-forms? 2. Does zero lie in the spectrum of the Laplacian acting on (1-forms on M)/Ker(d)?