Norms on the cohomology of hyperbolic 3-manifolds

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, 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.

29 pages, 2 figures. V2: Added Section 5.3 on related work of Kronheimer and Mrowka. V3: Minor improvements. V4: To appear in Inventiones Mathematicae V5: Theorem and equation numbering changed to match published version

References in corpus (2)

• Torsion homology growth and cycle complexity of arithmetic manifolds
• Bounded combinatorics and uniform models for hyperbolic 3-manifolds

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• Cubulating Surface-by-free Groups
• Monopole Floer homology and the spectral geometry of three-manifolds
• Harmonic Forms, Minimal Surfaces and Norms on Cohomology of Hyperbolic $3$-Manifolds
• Geometry of the smallest 1-form Laplacian eigenvalue on hyperbolic manifolds
• Homological norms on nonpositively curved manifolds