On Structure Groups of Set-Theoretic Solutions to the Yang–Baxter Equation

This paper explores the structure groups G(X,r) of finite non-degenerate set-theoretic solutions (X,r) to the Yang–Baxter equation. Namely, we construct a finite quotient $\overline {G}_{(X,r)}$ of G(X,r), generalizing the Coxeter-like groups introduced by Dehornoy for involutive solutions. This yields a finitary setting for testing injectivity: if X injects into G(X,r), then it also injects into $\overline {G}_{(X,r)}$. We shrink every solution to an injective one with the same structure group, and compute the rank of the abelianization of G(X,r). We show that multipermutation solutions are the only involutive solutions with diffuse structure groups; that only free abelian structure groups are bi-orderable; and that for the structure group of a self-distributive solution, the following conditions are equivalent: bi-orderable, left-orderable, abelian, free abelian and torsion free.