Stable commutator length on free $\mathbb{Q}$-groups
Francesco Fournier-Facio
TL;DR
This work studies stable commutator length (scl) in free Q-groups, proving every non-trivial element has positive scl and that the natural free-group embedding into its Q-completion is isometric for scl. It develops the framework of rational extensions and A-groups, showing scl is preserved under these constructions and that free or hyperbolic groups yield infinite-dimensional spaces of homogeneous quasimorphisms modulo homomorphisms. By leveraging Bavard duality, it connects scl positivity to abundant quasimorphisms and demonstrates that non-trivial positive theory arises in Q-groups and their completions. The paper further links scl rationality to surface-group rationality by embedding non-orientable surface groups into free Q-groups, highlighting deep connections between scl, bounded cohomology, and long-standing open problems in topology.
Abstract
We study stable commutator length on free $\mathbb{Q}$-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 $\mathbb{Q}$-group has an infinite-dimensional space of homogeneous quasimorphisms modulo homomorphisms, answering a question of Casals-Ruiz, Garreta, and de la Nuez Gonz{á}lez. We conjecture that stable commutator length is rational on free $\mathbb{Q}$-groups. This is connected to the long-standing problem of rationality on surface groups: indeed, we show that free $\mathbb{Q}$-groups contain isometrically embedded copies of non-orientable surface groups.