TITLE: Second bounded cohomology and normal subgroups
QUESTION [5 upvotes]: It may be a naive question, but:

If a finitely generated group has an infinite-dimensional second bounded cohomology group, does it imply that it contains "many" normal subgroups?

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?

REPLY [2 votes]: I finally found the answer to my question.

Proposition: There exists a finitely presented simple group whose second bounded cohomology group is infinite-dimensional.

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. On the other hand, Caprace and Fujiwara, in their article Rank one isometries of buildings and quasi-morphisms of Kac-Moody groups, 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).