Homogeneous quasimorphism

Suppose $G$ is a group. A homogeneous quasimorphism on $G$ is a quasimorphism on $G$ that is also a 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 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 1-homomorphism of groups condition: For any $x \in G$ and $n \in \mathbb{Z}$, we have $\mu(x^n) = (\mu(x))^n$.
• 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.