arXiv subjects

Igor Haladjian

Publications and source records attributed to Igor Haladjian.

4 recordsLinked to original sources

arXiv2025

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.

math.GR↗

arXiv2025

J-braid groups are torus necklace groups

We construct a family of links we call torus necklaces for which the link groups are precisely the braid groups of generalised $J$-reflection groups. Moreover, this correspondence exhibits the meridians of the aforementioned link groups as braid reflections. In particular, this construction generalises to all irreducible rank two complex reflection groups a well-known correspondence between some rank two complex braid groups and some torus knot groups. In addition, as abstract groups, we show that the family of link groups associated to Seifert links coincides with the family of circular groups. This shows that every time a link group has a non-trivial center, it is a Garside group.

math.GT↗

arXiv2024

Braid groups of J-reflection groups and associated classical and dual Garside structures

The family of $J$-reflection groups can be seen as a combinatorial generalisation of irreducible rank two complex reflection groups and was introduced by the author in a previous article. In this article, we define the braid groups associated to $J$-reflection groups, which coincide with the complex braid group when the $J$-reflection group is finite. We show that the isomorphism type of the braid groups only depend on the reflection isomorphism types of the corresponding $J$-reflection groups. Moreover, we show that these braid groups are always abstractly isomorphic to circular groups. At the same time, we show that the center of the braid groups is cyclic and sent onto the center of the corresponding $J$-reflection groups under the natural quotient. Finally, we exhibit two Garside structures for each braid group of $J$-reflection group. These structures generalise the classical and dual Garside structures (when defined) of rank two irreducible complex reflection groups. In particular, the dual Garside structure of $J$-reflection groups provides candidates for dual monoids associated to the irreducible complex reflection groups of rank two which do not already have one.

math.GR↗

arXiv2024

A Combinatorial Generalisation of Rank two Complex Reflection Groups via Generators and Relations

Complex reflection groups of rank two are precisely the finite groups in the family of groups that we call J-reflection groups. These groups are particular cases of J-groups as defined by Achar & Aubert in 2008. The family of J-reflection groups generalises both complex reflection groups of rank two and toric reflection groups, a family of groups defined and studied by Gobet. We give uniform presentations by generators and relations of J-reflection groups, which coincide with the presentations given by Brou\'e, Malle and Rouquier when the groups are finite. In particular, these presentations provide uniform presentations for complex reflection groups of rank two where the generators are reflections (however the proof uses the classification of irreducible complex reflection groups). Moreover, we show that the center of J-reflection groups is cyclic, generalising what happens for irreducible complex reflection groups of rank two and toric reflection groups. Finally, we classify J-reflection groups up to reflection isomorphisms.

math.GR↗