Scl, sails and surgery

TL;DR

This paper links stable commutator length in free groups to the geometry of sails in polyhedral cones, revealing piecewise rational linearity, computability via integer programming, and complex spectral properties.

Contribution

It establishes a novel connection between scl and polyhedral geometry, enabling new computational methods and spectral analysis in free groups.

Findings

scl norm is piecewise rational linear in free products of Abelian groups

scl can be computed via integer programming

scl spectrum contains elements with dense rational congruences and well-ordered sequences

Abstract

We establish a close connection between stable commutator length in free groups and the geometry of sails (roughly, the boundary of the convex hull of the set of integer lattice points) in integral polyhedral cones. This connection allows us to show that the scl norm is piecewise rational linear in free products of Abelian groups, and that it can be computed via integer programming. Furthermore, we show that the scl spectrum of nonabelian free groups contains elements congruent to every rational number modulo $\mathbb{Z}$, and contains well-ordered sequences of values with ordinal type $\omega^\omega$. Finally, we study families of elements $w(p)$ in free groups obtained by surgery on a fixed element $w$ in a free product of Abelian groups of higher rank, and show that $\scl(w(p)) \to \scl(w)$ as $p \to \infty$.