Stable commutator length on free Q‐groups
Abstract
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.
Fields of Study & Subjects
Geometric and Algebraic Topology
Geometric Analysis and Curvature Flows
Mathematical Dynamics and Fractals
Publication details
- Journal / source
- Bulletin of the London Mathematical Society
- Volume
- 58
- Issue
- 1
- Publication date
- 1 January 2026
- Language
- EN