Suppose that $G$ is a group and $d$ is a left-invaraint metric on $G$, e.g., the word metric (provided that $G$ is finitely-generated) or distance function determined by a left-invariant Riemannian metric on $G$ (provided that $G$ is a connected Lie group). Let $\Gamma$ be another group.

A map $f: \Gamma\to G$ is called a *quasi-homomorphism* if there exists a constant $A$ so that for all $\gamma_1, \gamma_2\in \Gamma$ we have
$$
d(f(\gamma_1 \gamma_2), f(\gamma_1) f(\gamma_2))\le A.
$$

There are two sources of unbounded quasi-homomorphisms I know:

Group homomorphisms (and their bounded perturbations).

Quasi-morphisms, i.e., quasi-homomorphisms to ${\mathbb R}$ (with the standard metric) or, equivalently, to ${\mathbb Z}$ (again, with the standard metric). (One can modify this construction by taking quasi-morphisms with values in other abelian groups, but I will not regard this as a separate source of examples.)

There is, by now, a considerable literature on constructing quasi-morphisms, both in geometric group theory and symplectic geometry.

One can combine 1 and 2, by taking, say, compositions and direct sums/products: I am considering this as, again a trivial modification.

Question 1: Are there are other sources of quasi-morphisms which do not reduce to 1 and 2?

If this is a bit vague, here is a sub-question:

Question 2: Suppose that $\Gamma$ is higher-rank (real rank $\ge 3$) irreducible lattice (thus, it has no unbounded quasimorphisms) and $G$ is a nonelementary hyperbolic group (to exclude homomorphisms $\Gamma\to G$ with infinite image). Are there examples of unbounded quasi-homomorphisms $\Gamma\to G$?

Is there any literature on this? I could not find anything, but maybe I was looking at wrong places.

Note that the setting of quasi-homomorphisms (in the case of nonabelian targets) is very different from the one of *quasi-actions*, in which case there is a substantial literature and I know most of it.