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Bounded cohomology of finitely generated Kleinian groups

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Abstract

Any action of a group \({\Gamma}\) on \({\mathbb{H}^3}\) by isometries yields a class in degree three bounded cohomology by pulling back the volume cocycle to \({\Gamma}\). We prove that the bounded cohomology of finitely generated Kleinian groups without parabolic elements distinguishes the asymptotic geometry of geometrically infinite ends of hyperbolic 3-manifolds. That is, if two homotopy equivalent hyperbolic manifolds with infinite volume and without parabolic cusps have different geometrically infinite end invariants, then they define a 2 dimensional subspace of bounded cohomology. Our techniques apply to classes of hyperbolic 3-manifolds that have sufficiently different end invariants, and we give explicit bases for vector subspaces whose dimension is uncountable. We also show that these bases are uniformly separated in pseudo-norm, extending results of Soma. The technical machinery of the Ending Lamination Theorem allows us to analyze the geometrically infinite ends of hyperbolic 3-manifolds with unbounded geometry.

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Acknowledgments

The author would like to thank Kenneth Bromberg for many hours of his time and for his patience. Helpful conversations with Maria Beatrice Pozzetti inspired the contents of Section 7.3, which were useful for improving a result from a previous draft of this manuscript. We would also like to thank the hospitality of the MSRI and partial support of the NSF under Grants DMS-1246989 and DMS-1440140.

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  1. Department of Mathematics, University of Utah, 155 South 1400 East, JWB 233, Salt Lake City, UT, 84112, USA

    James Farre

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  1. James Farre
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Farre, J. Bounded cohomology of finitely generated Kleinian groups. Geom. Funct. Anal. 28, 1597–1640 (2018). https://doi.org/10.1007/s00039-018-0470-y

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