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.