Stable commutator length is rational in free groups

Abstract

For any group, there is a natural (pseudo-)norm on the vector space B1 of real (group) 1-boundaries, called the stable commutator length norm. This norm is closely related to, and can be thought of as a relative version of, the Gromov (pseudo)-norm on (ordinary) homology. We show that for a free group, the unit ball of this pseudo-norm is a rational polyhedron. It follows that stable commutator length in free groups takes on only rational values. Moreover every element of the commutator subgroup of a free group rationally bounds an injective map of a surface group. The proof of these facts yields an algorithm to compute stable commutator length in free groups. Using this algorithm, we answer a well-known question of Bavard in the negative, constructing explicit examples of elements in free groups whose stable commutator length is not a half-integer.

21 pages, 4 figures; version 2 incorporates referees' suggestions

References in corpus (1)

• Faces of the scl norm ball

Cited by in corpus (9)

• Matrix Group Integrals, Surfaces, and Mapping Class Groups I: $U(n)$
• Word Measures on Symmetric Groups
• Surface subgroups from linear programming
• Scl in graphs of groups
• Scl in free products
• The Minimal Genus of Homology Classes in a Finite 2-Complex
• Simplicial volume of one-relator groups and stable commutator length
• On stable commutator length of non-filling curves in surfaces
• Stable immersions in orbifolds