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Quasihomomorphisms to abelian groups are an important tool in geometric group theory and symplectic geometry. In this talk, I will discuss homomorphisms with discrete noncommutative targets. It turns out that such quasihomomorphisms are "rigid" in the sense that they are all obtained from homomorphisms to groups and quasihomomorphisms with abelian target groups via several functorial constructions. This is a joint work with Koji Fujiwara.