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

TL;DR

The paper develops a comprehensive framework linking generalized J-groups to rank-2 complex reflection groups, J-braid groups, and Seifert link groups. By introducing J(K) and J(binomial{K}{K'}), it extends the classical J-groups and characterizes when these generalized groups are finite or finitely generated, tying finite-type cases to rank-2 complex reflection groups. It then constructs and analyzes torsion quotients of J-braid groups, showing they are reflection isomorphic to finitely generated generalized J-groups, and proves a two-way correspondence: every finitely generated generalized J-group is a torsion quotient of a J-braid group. The work culminates in a complete classification up to reflection isomorphism, including reduced J-groups, and connects these algebraic objects to Seifert link groups, proving that finite torsion quotients of Seifert links recover the rank-2 complex reflection groups, with torsion quotients detecting link isotopy in this setting.

Abstract

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.

Figures (2)

• Figure 1: The torus necklaces $\mathrm L_*^*(3,4)$, $\mathrm L_*(3,4)$, $\mathrm L^*(3,4)$ and $\mathrm L(3,4)$
• Figure 2: The keychain $\mathrm K(5)$

Theorems & Definitions (178)

• Theorem 1.1
• Definition 1.2: Generalized $J$-groups
• Theorem 1.3: Proposition \ref{['prop:finite-type_=_finiment_engendre']} and Corollary \ref{['cor:finite_generalised_j_groups']}
• Corollary 1.4: Corollary \ref{['cor:finite_generalized_j_groups_are_crgs']}
• Proposition 1.5: Proposition \ref{['order_center_generalized_j_groups']}
• Corollary 1.6: Corollary \ref{['cor:torsion_generalized_j_groups']}
• Theorem 1.7: Corollary \ref{['cor:embedding_torsion_quotients']}
• Theorem 1.8: Theorem \ref{['theo:generalized_j_groups_torsion_quotients']}
• Proposition 1.9
• Proposition 1.10