quasi-homomorphisms of groups

Question

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.

Answer 1

As it turned out (contrary to my expectations), all quasihomomorphisms with noncommutative target groups (which are discrete with proper left invariant metric) are obtained via some natural constructions starting from homomorphisms to groups and quasihomomorphisms to abelian groups:

This is the main result of our paper with Koji Fujiwara here.

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.

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Update. Many things have happened in the last 9 years. In particular, very recently it turned out that the answer is quite different in the case when the target group is a connected noncompact Lie group and the group $\Gamma$ is the fundamental group of a compact negatively curved manifold. Namely, a large supply of quasihomomorphisms (much bigger than in our paper with Fujiwara) was discovered in

Michael Brandenbursky, Misha Verbitsky, Non-commutative Barge-Ghys quasimorphisms, arXiv:2212.12958, 2022.