Homogeneous quasimorphism

Definition
Suppose $$G$$ is a group. A homogeneous quasimorphism on $$G$$ is a defining ingredient::quasimorphism on $$G$$ that is also a conjunction involving::1-homomorphism of groups. Explicitly, a function $$\mu:G \to \R$$ is termed a homogeneous quasimorphism (sometimes abbreviated as hqm) if and only if it satisfies both these conditions:


 * 1) The conjunction involving::quasimorphism condition: There exists a positive real number $$D$$ such that $$|\mu(xy) - \mu(x) - \mu(y)| \le D$$ for all $$x,y \in G$$. Note that $$D$$ depends on $$\mu$$ but not on the choice of $$x$$ or $$y$$. The smallest value of $$D$$ that works is termed the defect of $$\mu$$ and is denoted $$D(\mu)$$.
 * 2) The conjunction involving::1-homomorphism of groups condition: For any $$x \in G$$ and $$n \in \mathbb{Z}$$, we have $$\mu(x^n) = (\mu(x))^n$$.

Facts

 * Given any quasimorphism, we can construct a homogeneous quasimorphism from it called its homogenization. For a quasimorphism $$f :G \to \R$$, the homogenization $$\mu_f$$ is defined as $$x \mapsto \lim_{n \to \infty} \frac{f(x^n)}{n}$$.
 * The collection of homogeneous quasimorphisms on a group $$G$$ is a vector space with pointwise addition and scalar multiplication of functions. This vector space is denoted $$Q(G)$$. The collection of all quasimorphisms (including the non-homogeneous ones) is a bigger normed vector space, and is denoted $$\hat(Q)(G)$$.
 * The quotient space $$Q(G)/\operatorname{Hom}(G,\R)$$ is a Banach space with the defect being the norm.