Quasimorphism

In group theory, given a group $$G$$, a quasimorphism (or quasi-morphism) is a function $$f:G\to\mathbb{R}$$ which is additive up to bounded error, i.e. there exists a constant $$D\geq 0$$ such that $$|f(gh)-f(g)-f(h)|\leq D$$ for all $$g, h\in G$$. The least positive value of $$D$$ for which this inequality is satisfied is called the defect of $$f$$, written as $$D(f)$$. For a group $$G$$, quasimorphisms form a subspace of the function space $$\mathbb{R}^G$$.

Examples

 * Group homomorphisms and bounded functions from $$G$$ to $$\mathbb{R}$$ 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 $$G=F_S$$ be a free group over a set $$S$$. For a reduced word $$w$$ in $$S$$, we first define the big counting function $$C_w:F_S\to \mathbb{Z}_{\geq 0}$$, which returns for $$g\in G$$ the number of copies of $$w$$ in the reduced representative of $$g$$. Similarly, we define the little counting function $$c_w:F_S\to\mathbb{Z}_{\geq 0}$$, returning the maximum number of non-overlapping copies in the reduced representative of $$g$$. For example, $$C_{aa}(aaaa)=3$$ and $$c_{aa}(aaaa)=2$$. Then, a big counting quasimorphism (resp. little counting quasimorphism) is a function of the form $$H_w(g)=C_w(g)-C_{w^{-1}}(g)$$ (resp. $$h_w(g)=c_w(g)-c_{w^{-1}}(g))$$.
 * The rotation number $$\text{rot}:\text{Homeo}^+(S^1)\to\mathbb{R}$$ is a quasimorphism, where $$\text{Homeo}^+(S^1)$$ denotes the orientation-preserving homeomorphisms of the circle.

Homogeneous
A quasimorphism is homogeneous if $$f(g^n)=nf(g)$$ for all $$g\in G, n\in \mathbb{Z}$$. It turns out the study of quasimorphisms can be reduced to the study of homogeneous quasimorphisms, as every quasimorphism $$f:G\to\mathbb{R}$$ is a bounded distance away from a unique homogeneous quasimorphism $$\overline{f}:G\to\mathbb{R}$$, given by :
 * $$\overline{f}(g)=\lim_{n\to\infty}\frac{f(g^n)}{n}$$.

A homogeneous quasimorphism $$f:G\to\mathbb{R}$$ has the following properties:
 * It is constant on conjugacy classes, i.e. $$f(g^{-1}hg)=f(h)$$ for all $$g, h\in G$$,
 * If $$G$$ is abelian, then $$f$$ 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 $$f:G\to\mathbb{Z}$$. In this case, the above discussion about homogeneous quasimorphisms does not hold anymore, as the limit $$\lim_{n\to\infty}f(g^n)/n$$ does not exist in $$\mathbb{Z}$$ in general.

For example, for $$\alpha\in\mathbb{R}$$, the map $$\mathbb{Z}\to\mathbb{Z}:n\mapsto\lfloor\alpha n\rfloor$$ is a quasimorphism. There is a construction of the real numbers as a quotient of quasimorphisms $$\mathbb{Z}\to\mathbb{Z}$$ by an appropriate equivalence relation, see Construction of the reals numbers from integers (Eudoxus reals).