Braid groups of 𝐽-reflection groups

The family of 𝐽-reflection groups form a combinatorial generalisation of irreducible rank-two complex reflection groups. 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 types 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 abstractly isomorphic to circular groups. At the same time, we show that the centre of these braid groups is cyclic and sent onto the centre of the corresponding 𝐽-reflection groups under the natural quotient.