Quasihomomorphisms with non-commutative target

Quasihomomorphisms (QHMs) are maps $f$ between groups such that the
homomorphic condition is boundedly satisfied. The case of QHMs with
abelian target is well studied and is useful for computing the second
bounded cohomology of groups. The case of target non-abelian has,
however, not been studied a lot.

We will see a technique for classifying QHMs $f: G \rightarrow H$ by Fujiwara and
Kapovich. We will give examples (sometimes with proofs!) for QHM in
various cases such as

• the image $H$  hyperbolic groups,
• the image $H$ discrete rank one isometries,
• the preimage $G$ cyclic / free group, etc.

Furthermore, we point out a relation between QHM and extensions by short
exact sequences.