Stable commutator length is rational in free groups

TL;DR

This paper proves that stable commutator length in free groups is rational, characterizes its unit ball as a rational polyhedron, and provides an algorithm for its computation, answering a longstanding question in the field.

Contribution

It establishes the rationality of stable commutator length in free groups, describes the geometric structure of its unit ball, and introduces an algorithm for calculating it.

Findings

The unit ball of the stable commutator length norm is a rational polyhedron.

Stable commutator length in free groups takes only rational values.

Explicit examples show stable commutator length can be non-half-integer.

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.