Quasimorphism

Definition
Suppose $$G$$ is a group. A quasihomomorphism on $$G$$ is a function $$f: G \to \R$$ (where $$\R$$ is the field of real numbers) satisfying the condition that there exists a positive real number $$D$$ such that for all $$x,y \in G$$, we have:

$$|f(xy) - f(x) - f(y)| \le D$$

Note that $$D$$ depends on $$f$$, but not on the choice of elements of $$G$$.

The smallest positive real number $$D$$ that works is called the defect of the quasimorphism $$f$$. A quasimorphism of defect 0 is the same as a homomorphism to $$\R$$.

Other names for this concept are quasihomomorphism (not to be confused with a different notion of quasihomomorphism of groups) and pseudocharacter.

Homogenization
A homogeneous quasimorphism is a quasiomorphism that is also a 1-homomorphism of groups, i.e., its restriction to any cyclic subgroup of $$G$$ is a homomorphism. For any quasimorphism $$f$$, we can consider its homogenization, defined as $$\mu_f := x \mapsto \lim_{n \to \infty} \frac{f(x^n)}{n}$$.

Facts

 * The collection of all quasimorphisms on a group $$G$$ is avector space, with the vector space structure being pointwise addition and scalar multiplication of functions. This vector space is denoted $$\hat{Q}(G)$$. The subspace of homogeneous quasimorphisms is denoted $$Q(G)$$. The quotient space $$\hat{Q}(G)/\operatorname{Hom}(G,\R)$$ is a normed vector space with the defect being the norm.

Definition in terms of the language of cohomology
Suppose $$G$$ is a group. Consider the cochain complex $$C^*(G;\R)$$ for the trivial group action of $$G$$ on $$\R$$. In particular:


 * $$C^1(G;\R)$$ is the additive group of all functions from $$G$$ to $$\R$$. In this case, $$C^1(G;\R)$$ has the structure of a $$\R$$-vector space.
 * $$C^2(G;\R)$$ is the additive group of all functions from $$G \times G$$ to $$\R$$, and also has the structure of a $$\R$$-vector space.

Consider the coboundary map of the cochain complex:

$$d_1: C^1(G;\R) \to C^2(G;R)$$

given as:

$$d_1(f) := (x,y) \mapsto f(x) + f(y) - f(xy)$$

The standard terminology is as follows:


 * The kernel of $$d_1$$ is the group of 1-cocycles for the trivial group action of $$G$$ on $$\R$$, which coincides with the group $$\operatorname{Hom}(G;\R)$$. Moreover, the group of 1-coboundaries is trivial, so this also coincides with the first cohomology group $$H^1(G;\R)$$. See first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms.
 * The image of $$d_1$$ is the group of 2-coboundaries for the trivial group action of $$G$$ on $$\R$$, and is denoted $$B^2(G;\R)$$. This, too, is a $$\R$$-vector space.
 * The vector space $$\hat{Q}(G)$$ that we are interested in is the inverse image under $$d_1$$ of the vector subspace $$B_{bdd}^2(G;\R)$$ of $$B^2(G;\R)$$ comprising the 2-coboundaries that are bounded maps to $$\R$$.
 * By the first isomorphism theorem he quotient space $$\hat{Q}(G)/\operatorname{Hom}(G,\R)$$ is isomorphic to $$B_{bdd}^2(G;\R)$$. Moreover, the defect norm on $$\hat{Q}(G)/\operatorname{Hom}(G,\R)$$ corresponds with the $$L^\infty$$-norm on this vector space.

Examples

 * Any set map from a group to $$\R$$ with a bounded image is a quasimorphism. In particular, any continuous map from a compact topological group to $$\R$$ is a quasimorphism. Examples include coordinate projections from compact manifolds embedded in $$\R^n$$. Note that the homogenization of any such quasimorphism is the zero quasimorphism, so such quasimorphisms are not interesting up to homogenization.
 * The rotation number quasimorphism is a homogeneous quasimorphism.