Scl, sails, and surgery

Abstract

We establish a close connection between the 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 programing. Furthermore, we show that the scl spectrum of non‐Abelian free groups contains elements congruent to every rational number modulo ℤ, and contains well‐ordered sequences of values with ordinal type ωω. 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)) → scl(w) as p → ∞.

Stable commutator length is rational in free groups

“…The program scallop (source code available at [8]) implements this algorithm and can be used to compute scl in 2 on words of length ∼ 60. Experiments using scallop reveal additional structure in the scl spectrum of a free group, partially explained in a forthcoming paper [7]. In particular, it is possible to prove rigorously that in any nonabelian free group, the image of scl contains nontrivial accumulation points and takes on values with any denominator.…”

Spectral gap of scl in free products

“…Calegari [3] shows how to compute scl in certain free products by a pair of linear programming problems, one for each of A and B.…”

Scl in graphs of groups

“…With the help of Lemma 4.15, there are several known methods of encoding to compute scl via linear programming [7,3,22,8,36]. In this section, we use an encoding similar to those in [8,19] to optimize our rationality result. We will use the notation from Section 4 and the setup in Lemma 4.15.…”