In group theory, given a group , a quasimorphism (or quasi-morphism) is a function which is additive up to bounded error, i.e. there exists a constant such that for all . The least positive value of for which this inequality is satisfied is called the defect of , written as . For a group , quasimorphisms form a subspace of the function space .
Examples Group homomorphisms and bounded functions from to are quasimorphisms. The sum of a group homomorphism and a bounded function is also a quasimorphism, and functions of this form are sometimes referred to as "trivial" quasimorphisms.Frigerio (2017), p. 12. Let be a free group over a set . For a reduced word in , we first define the big counting function , which returns for the number of copies of in the reduced representative of . Similarly, we define the little counting function , returning the maximum number of non-overlapping copies in the reduced representative of . For example, and . Then, a big counting quasimorphism (resp. little counting quasimorphism) is a function of the form (resp. . The rotation number is a quasimorphism, where denotes the orientation-preserving homeomorphisms of the circle, and denotes its universal covering.
Homogeneous A quasimorphism is homogeneous if for all . It turns out the study of quasimorphisms can be reduced to the study of homogeneous quasimorphisms, as every quasimorphism is a bounded distance away from a unique homogeneous quasimorphism , given by : . A homogeneous quasimorphism has the following properties: It is constant on conjugacy classes, i.e. for all , If is abelian, then is a group homomorphism. The above remark implies that in this case all quasimorphisms are "trivial".
Integer-valued One can also define quasimorphisms similarly in the case of a function . In this case, the above discussion about homogeneous quasimorphisms does not hold anymore, as the limit does not exist in in general.
For example, for , the map is a quasimorphism. There is a construction of the real numbers as a quotient of quasimorphisms by an appropriate equivalence relation, see Construction of the reals numbers from integers (Eudoxus reals).
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Further reading What is a Quasi-morphism? by D. Kotschick