Generalized J-groups, J-braid groups and Seifert link groups

The family of J-groups was introduced by Achar and Aubert with the goal of providing Coxeter-like combinatorial tools for studying rank 2 complex reflection groups. However, J-groups lack an explicit presentation with abstract reflections as generators. This gap was filled by Gobet, and later by the second author, for the subfamily of so-called J-reflection groups. The obtained presentations then gave rise to a concept of J-braid group, which coincides with the link groups of torus necklaces. In this paper we study a generalization of J-groups. We determine which of these groups are finitely generated. We show that, as for classical J-groups, the family of finite generalized J-groups coincides with the family of rank 2 complex reflection groups. We also show that finitely generated generalized J-groups coincide with what we call the torsion quotients of J-braid groups. We deduce explicit presentations for all finitely generated generalized J-groups, where the generators are abstract reflections. We also complete the classification of these groups up to reflection isomorphism. As a byproduct of these results, we obtain that a quotient of a Seifert link group obtained by adding torsion to meridians somehow determines the link up to isotopy. Moreover, such a quotient is finite if and only if it is isomorphic to a complex reflection group of rank two.