Norms on the cohomology of hyperbolic 3-manifolds

Jeffrey F. Brock, Nathan M. Dunfield

Published 2015-10-21Version 1

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, 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.