21.48 (2026)

A quasimorphism on a group $G$ is a function $f : G \to \mathbb{R}$ such that the quantity $\sup_{g,h} |f(g) + f(h) - f(gh)|$ is finite. A quasimorphism is homogeneous if it restricts to a homework on every cyclic subgroup of $G$.

Let $G$ be a group admitting an unbounded homogeneous quasimorphism $G \to \mathbb{R}$ that is not a homomorphism. Must $G$ contain a non-abelian free subgroup?