Coxeter-like groups for set-theoretic solutions of the Yang-Baxter equation. (Q367175)

Let \(R\colon X\times X\to X\times X\) be a non-degenerate involutive set-theoretic solution of the Yang-Baxter equation. \textit{P. Etingof, T. Schedler} and \textit{A. Soloviev} [Duke Math. J. 100, No. 2, 169-209 (1999; Zbl 0969.81030)] associated a structure group \(G(R)\) to \(R\), and Chouraqui recently made the fundamental observation that \(G(R)\) is a Garside group. Prominent examples of Garside groups are the Artin braid groups.NEWLINENEWLINE It is an open problem whether there is an analogue of the Coxeter group for every Garside group. Using the reviewer's translation of solutions \(R\) (as above) into certain sets \(X\) with a binary operation (cycle sets) and the fact that every cycle set is equivalent to a group of I-type, the author obtains a positive answer to the problem for Garside groups arising as structure groups \(G(R)\).