Homogenization of a quasimorphism

Definition

Suppose $f:G\to \mathbb {R}$ is a quasimorphism. The homogenization of $f$ is a homogeneous quasimorphism $\mu _{f}$ (sometimes denoted ${\overline {f}}$) defined as follows:

$\mu _{f}:=x\mapsto \lim _{n\to \infty }{\frac {f(x^{n})}{n}}$

Facts

Defect

• Upper bound: The defect $D(\mu _{f})$ of the homogenization is at most twice the defect $D(f)$ of $f$.
• Lower bound: There is no effective lower bound besides the obvious one of zero. To see this, note that any quasimorphism with bounded image has homogenization the zero map, even though we can have quasimorphisms with bounded image that have arbitrarily large defect.