Quasimorphism
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. 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. 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). Notes

References

Further reading 
What is a Quasi-morphism? by D. Kotschick

Group theory
Additive functions