Involutive Yang-Baxter groups

In 1992 Drinfeld posed the question of finding the set-theoretic solutions of the Yang-Baxter equation. Recently, Gateva-Ivanova and Van den Bergh and Etingof, Schedler and Soloviev have shown a group-theoretical interpretation of involutive non-degenerate solutions. Namely, there is a one-to-one correspondence between involutive non-degenerate solutions on finite sets and groups of I I -type. A group G G of I I -type is a group isomorphic to a subgroup of F a n ⋊ S y m n Fa_n Sym_n so that the projection onto the first component is a bijective map, where F a n Fa_n is the free abelian group of rank n n and S y m n Symₙ is the symmetric group of degree n n . The projection of G G onto the second component S y m n Sym_n we call an involutive Yang-Baxter group (IYB group). This suggests the following strategy to attack Drinfeld’s problem for involutive non-degenerate set-theoretic solutions. First classify the IYB groups and second, for a given IYB group G G , classify the groups of I I -type with G G as associated IYB group. It is known that every IYB group is solvable. In this paper some results supporting the converse o