Quasimorphism

Summary

In group theory, given a group $G$, a quasimorphism (or quasi-morphism) is a function $f:G\to\mathbb{R}$ which is additive up to bounded error, i.e. there exists a constant $D\geq 0$ such that $|f(gh)-f(g)-f(h)|\leq D$ for all $g, h\in G$. The least positive value of $D$ for which this inequality is satisfied is called the defect of $f$, written as $D(f)$. For a group $G$, quasimorphisms form a subspace of the function space $\mathbb{R}^G$.