# Groups for which quasimorphisms are close to homomorphisms

Let $$GG$$ be a group. A real-valued map $$φ:G→R\varphi:G \to \mathbb{R}$$ is called a quasimorphism if it’s a group homomorphism up to a uniformly bounded error:
$$sup\sup_{g,h\in G}|\varphi(gh)-\varphi(g)-\varphi(h)|<\infty.$$

In this question,
the OP points out that if the comparison homomorphism
$$c : H_b^2(G,\mathbb{R}) \to H^2(G,\mathbb{R})c : H_b^2(G,\mathbb{R}) \to H^2(G,\mathbb{R})$$
on the second bounded cohomomology group has trivial kernel, then all quasimorphisms are bounded perturbations of homomorphisms, i.e for each quasimorphism $$\varphi\varphi$$, there is a homomorphism within uniformly bounded distance of $$\varphi\varphi$$.

Is there a name for groups with this property, i.e. groups for which $$\text{ker}(c)\text{ker}(c)$$ is trivial? In particular I'm looking for examples of (non-elementary) Gromov-hyperbolic groups with this property - does anyone know of such groups?

I cannot comment, so I'll put it as an answer, which also summarizes the comments.

In the literature, groups with trivial comparison map are sometimes referred to as groups with vanishing stable commutator length (scl), or groups without nontrivial quasimorphisms. The characterization via scl follows from Bavard's duality.

Danny Calegari's book is an extensive introduction to scl, which includes discussions of groups with no or few nontrivial quasimorphisms. Here are some examples (all section numbers refer to the book):

(1) amenable groups (Section 2.4);

(2) lattices of higher rank irreducible semisimple Lie groups (Section 5.2), this is mentioned by tessellation in the comment;

(3) uniformly perfect groups (since they have trivial scl), which holds for many transformation groups, typically because of the suspension trick. This includes for example $$\mathrm{Homeo}^+(S^1)\mathrm{Homeo}^+(S^1)$$ (Section 2.3) and the mapping class group of the sphere minus a Cantor set (see this blog post)

(4) some other transformation groups, including all subgroups of $$\mathrm{PL}^{+}(I)\mathrm{PL}^{+}(I)$$ (see this paper), some subgroups of $$\mathrm{Homeo}^+(S^1)\mathrm{Homeo}^+(S^1)$$ such as the Thompson group $$TT$$ (Section 5.2).

(5) groups that obey a law (see this paper).

As is mentioned by YCor, non-elementary hyperbolic groups are on the opposite side: they have a huge number of nontrivial quasimorphisms (with continuum dimension) and a rich theory of scl (see Epstein and Fujiwara as well as Chapter 3 of the book above).

It is also worth mentioning that mapping class groups of closed hyperbolic surfaces also have infinite-dimensional nontrivial quasimorphisms (Bestvina and Fujiwara), and the same holds for the mapping class group of the plane minus a Cantor set (by Juliette Bavard).

In general, one should expect groups with nice actions on non-positively curved spaces to have many nontrivial quasimorphisms.