quasi-homomorphisms of groups

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:

1. Group homomorphisms (and their bounded perturbations).

2. 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.

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For question 2, note that a hyperbolic group $G$ has lots of quasi-morphisms, so any quasi-homomorphism $\Gamma\to G$ would have to lie in the "kernel" of all of these, which gives evidence to me that its image should be finite. –  Ian Agol Jul 17 '13 at 5:05
Are you aware of Ozawa's work arxiv.org/abs/0911.3975? –  Mikael de la Salle Jul 17 '13 at 5:27
See also the construction of Rolli arxiv.org/abs/0911.4234. –  Andreas Thom Jul 17 '13 at 6:38
@MikaeldelaSalle: Thank you, I did not know about it. In particular, Ozawa proves that any quasi-homomorphism $SL(n,Z)\to G$ with $G$ amenable or hyperbolic, has bounded image, which supports the guess that answer to Question 2 is negative. –  Misha Jul 17 '13 at 11:59
@AndreasThom: Thank you, I did not know of the construction. Sadly, however, Rolli needs a bi-invariant metric on the target group. The only interesting (nonabelian) examples I know are groups of Hamiltonian symplectomorphisms with Hofer metric. –  Misha Jul 17 '13 at 12:02

To the list of constructions I had in my question one would have to add extension from a finite index subgroup in $\Gamma$ and a lifting construction using a bounded cohomology class in $H^2(\Gamma, A)$, where $A$ is an abelian group. In particular, the answer to Question 2 is negative.