Inventiones mathematicae

, Volume 210, Issue 2, pp 531–558 | Cite as

Norms on the cohomology of hyperbolic 3-manifolds

  • Jeffrey F. Brock
  • Nathan M. Dunfield


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, Şengün, 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.



Brock was partially supported by US NSF Grant DMS-1207572 and the GEAR Network (US NSF Grant DMS-1107452), and this work was partially conducted at ICERM. Dunfield was partially supported by US NSF Grant DMS-1106476, the GEAR Network, a Simons Fellowship, and this work was partially completed while visiting ICERM, the University of Melbourne, and the Institute for Advanced Study. We gratefully thank Nicolas Bergeron, Haluk Şengün, Akshay Venkatesh, Anil Hirani, Mike Freedman, Tom Church, Tom Mrowka, and Peter Kronheimer for helpful conversations and comments. Finally, we thank the several referees for their helpful comments and suggestions.


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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsBrown UniversityProvidenceUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

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