The cup product of Brooks quasimorphisms
-
Michelle Bucher
Abstract
We prove the vanishing of the cup product of the bounded cohomology classes associated to any two Brooks quasimorphisms on the free group. This is a consequence of the vanishing of the square of a universal class for tree automorphism groups.
Acknowledgements
The authors are grateful to Tobias Hartnick for his comments.
References
[1] R. Brooks, Some remarks on bounded cohomology, Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, Ann. of Math. Stud. 97, Princeton University Press, Princeton (1981), 53–63. 10.1515/9781400881550-006Search in Google Scholar
[2]
M. Bucher and N. Monod,
The bounded cohomology of
[3] B. Duchesne and N. Monod, Group actions on dendrites and curves, preprint (2016), https://arxiv.org/abs/1609.00303v2. 10.5802/aif.3209Search in Google Scholar
[4] R. I. Grigorchuk, Some results on bounded cohomology, Combinatorial and Geometric Group Theory (Edinburgh 1993), London Math. Soc. Lecture Note Ser. 204, Cambridge University Press, Cambridge (1995), 111–163. 10.1017/CBO9780511566073.011Search in Google Scholar
[5] T. Hartnick and P. Schweitzer, On quasioutomorphism groups of free groups and their transitivity properties, J. Algebra 450 (2016), 242–281. 10.1016/j.jalgebra.2015.10.010Search in Google Scholar
[6] N. Heuer, Cup product in bounded cohomology of the free group, preprint (2017), https://arxiv.org/abs/1710.03193. 10.2422/2036-2145.201802_007Search in Google Scholar
[7] B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972), 1–96. 10.1090/memo/0127Search in Google Scholar
[8]
Y. Mitsumatsu,
Bounded cohomology and
[9] N. Monod, Continuous Bounded Cohomology of Locally Compact Groups, Lecture Notes in Math. 1758, Springer, Berlin, 2001. 10.1007/b80626Search in Google Scholar
[10] N. Monod and Y. Shalom, Negative curvature from a cohomological viewpoint and cocycle superrigidity, C. R. Math. Acad. Sci. Paris 337 (2003), no. 10, 635–638. 10.1016/j.crma.2003.10.002Search in Google Scholar
[11] N. Monod and Y. Shalom, Cocycle superrigidity and bounded cohomology for negatively curved spaces, J. Differential Geom. 67 (2004), no. 3, 395–455. 10.4310/jdg/1102091355Search in Google Scholar
[12] A. H. Rhemtulla, A problem of bounded expressibility in free products, Proc. Camb. Philos. Soc. 64 (1968), 573–584. 10.1017/S0305004100043231Search in Google Scholar
[13] R. A. Ryan, Introduction to Tensor Products of Banach Spaces, Springer Monogr. Math., Springer, London, 2002. 10.1007/978-1-4471-3903-4Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Upper bounds for geodesic periods over rank one locally symmetric spaces
- Additive bases with coefficients of newforms
- A note on split extensions of bialgebras
- Petersson norm of cusp forms associated to real quadratic fields
- A realization theorem for sets of lengths in numerical monoids
- On the non-existence of the Mackey topology for locally quasi-convex groups
- The Davies method revisited for heat kernel upper bounds of regular Dirichlet forms on metric measure spaces
- The cup product of Brooks quasimorphisms
- Heat kernel estimates for time fractional equations
- Extreme non-Arens regularity of the group algebra
- Rational homology and homotopy of high-dimensional string links
- Global integrability for solutions to some anisotropic problem with nonstandard growth
- Hardy operators on Musielak–Orlicz spaces
- Contractibility of the stability manifold for silting-discrete algebras
- Order of the canonical vector bundle over configuration spaces of spheres
- An endpoint version of uniform Sobolev inequalities
- Space-time L2 estimates, regularity and almost global existence for elastic waves
- k-spaces and duals of non-archimedean metrizable locally convex spaces
- Path homology theory of multigraphs and quivers
Articles in the same Issue
- Frontmatter
- Upper bounds for geodesic periods over rank one locally symmetric spaces
- Additive bases with coefficients of newforms
- A note on split extensions of bialgebras
- Petersson norm of cusp forms associated to real quadratic fields
- A realization theorem for sets of lengths in numerical monoids
- On the non-existence of the Mackey topology for locally quasi-convex groups
- The Davies method revisited for heat kernel upper bounds of regular Dirichlet forms on metric measure spaces
- The cup product of Brooks quasimorphisms
- Heat kernel estimates for time fractional equations
- Extreme non-Arens regularity of the group algebra
- Rational homology and homotopy of high-dimensional string links
- Global integrability for solutions to some anisotropic problem with nonstandard growth
- Hardy operators on Musielak–Orlicz spaces
- Contractibility of the stability manifold for silting-discrete algebras
- Order of the canonical vector bundle over configuration spaces of spheres
- An endpoint version of uniform Sobolev inequalities
- Space-time L2 estimates, regularity and almost global existence for elastic waves
- k-spaces and duals of non-archimedean metrizable locally convex spaces
- Path homology theory of multigraphs and quivers
