It may be a naive question, but:



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> If a finitely generated group has an infinite-dimensional second bounded cohomology group, does it imply that it contains "many" normal subgroups?
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But "many", typically I have in mind "infinitely many" or even "uncountably many".


I am not familiar with bounded cohomology, but several articles deduce that the second bounded cohomology group is "big" from suitable actions on Gromov-hyperbolic spaces, where similar (but stronger) conditions can be used to construct normal subgroups via small cancellation. So I am wondering if there exists a connection between having a big second bounded cohomology and having many normal subgroups. For instance, can a simple group have an infinite-dimensional second bounded cohomology group?



Second bounded cohomology and normal subgroups
I finally found the answer to my question.



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> **Proposition:** There exists a finitely presented simple group whose second bounded cohomology group is infinite-dimensional.
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Such an example comes from the study of Kac-Moody groups. The simplicity of such groups is studied in Caprace and Rémy's article [Simplicity and superrigidity of twin building lattices](https://arxiv.org/abs/math/0607664). On the other hand, Caprace and Fujiwara, in their article [Rank one isometries of buildings and quasi-morphisms of Kac-Moody groups](https://arxiv.org/abs/0809.0470), showed that many Kac-Moody groups act on a building with rank-one isometries, and, following a previous work of Bestvina and Fujiwara, they constructed "many" quasi-morphisms, proving that $\widetilde{QH}$ is infinite-dimensional. As $\widetilde{QH}$ naturally lives inside $H\_b^2$, the desired conclusion follows.


As an interesting consequence, it follows that such Kac-Moody groups are simple but not uniformly perfect (and so, not uniformly simple). 


