Norms on the cohomology of hyperbolic 3-manifolds

TL;DR

This paper explores the relationship between the Thurston and harmonic norms on the first cohomology of hyperbolic 3-manifolds, establishing proportionality with explicit constants and analyzing their sharpness.

Contribution

It refines existing bounds between these norms, providing explicit constants depending on volume and injectivity radius, and presents examples illustrating the sharpness of these estimates.

Findings

Norms are roughly proportional with explicit constants

Examples show some estimates are sharp

Thurston norm can grow exponentially with volume

Abstract

We study the relationship between two norms on the first cohomology of a hyperbolic 3-manifold: the purely topological Thurston norm and the more geometric harmonic norm. Refining recent results of Bergeron, \c{S}eng\"un, and Venkatesh as well as older work of Kronheimer and Mrowka, we show that these norms are roughly proportional with explicit constants depending only on the volume and injectivity radius of the hyperbolic 3-manifold itself. Moreover, we give families of examples showing that some (but not all) qualitative aspects of our estimates are sharp. Finally, we exhibit closed hyperbolic 3-manifolds where the Thurston norm grows exponentially in terms of the volume and yet there is a uniform lower bound on the injectivity radius.