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Commentarii Mathematici Helvetici

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Volume 94, Issue 1, 2019, pp. 89–139
DOI: 10.4171/CMH/456

Published online: 2019-03-05

The zero norm subspace of bounded cohomology of acylindrically hyperbolic groups

Federico Franceschini[1], Roberto Frigerio[2], Maria Beatrice Pozzetti[3] and Alessandro Sisto[4]

(1) Karlsruher Institut für Technologie (KIT), Karlsruhe, Germany
(2) Università di Pisa, Italy
(3) Universität Heidelberg, Germany
(4) ETH Zürich, Switzerland

We construct combinatorial volume forms of hyperbolic three manifolds fibering over the circle. These forms define non-trivial classes in bounded cohomology. After introducing a new seminorm on exact bounded cohomology, we use these combinatorial classes to show that, in degree 3, the zero norm subspace of the bounded cohomology of an acylindrically hyperbolic group is infinite dimensional. In an appendix we use the same techniques to give a cohomological proof of a lower bound, originally due to Brock, on the volume of the mapping torus of a cobounded pseudo-Anosov homeomorphism of a closed surface in terms of its Teichmüller translation distance.

Keywords: Hyperbolic manifolds, mapping torus, relatively hyperbolic group, pseudo- Anosov automorphism, simplicial volume, Riemannian volume, quasi-cocycles, homological bicombing

Franceschini Federico, Frigerio Roberto, Pozzetti Maria Beatrice, Sisto Alessandro: The zero norm subspace of bounded cohomology of acylindrically hyperbolic groups. Comment. Math. Helv. 94 (2019), 89-139. doi: 10.4171/CMH/456