Abstract We study stable commutator length on free ‐groups. We prove that every non‐identity element has positive stable commutator length, and that the corresponding free group embeds isometrically. We deduce that a non‐abelian free ‐group has an infinite‐dimensional space of homogeneous quasimorphisms modulo homomorphisms, answering a question of Casals–Ruiz, Garreta and de la Nuez González. We conjecture that stable commutator length is rational on free ‐groups. This is connected to the long‐standing problem of rationality on surface groups: indeed, we show that free ‐groups contain isometrically embedded copies of non‐orientable surface groups.
Francesco Fournier‐Facio (Thu,) studied this question.