Homogeneous quasimorphism

Definition

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 $\mu :G\to \mathbb {R}$ 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 $|\mu (xy)-\mu (x)-\mu (y)|\leq D$ 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 $\mu (x^{n})=(\mu (x))^{n}$.

Facts

• Given any quasimorphism, we can construct a homogeneous quasimorphism from it called its homogenization. For a quasimorphism $f:G\to \mathbb {R}$, 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 $Q(G)/\operatorname {Hom} (G,\mathbb {R} )$ is a Banach space with the defect being the norm.