TORIC REFLECTION GROUPS

Several finite complex reflection groups have a braid group that is isomorphic to a torus knot group. Thereflection group is obtained from the torus knot group by declaring meridians to have orderkfor somek >= 2, and meridians are mapped to reflections. We study all possible quotients of torus knot groupsobtained by requiring meridians to have finite order. Using the theory ofJ-groups of Achar and Aubert['On rank 2 complex reflection groups',Comm. Algebra36(6) (2008), 2092-2132], we show that thesegroups behave like (in general, infinite) complex reflection groups of rank two. The large family of 'toricreflection groups' that we obtain includes, among others, all finite complex reflection groups of rank twowith a single conjugacy class of reflecting hyperplanes, as well as Coxeter's truncations of the 3-strandbraid group. We classify these toric reflection groups and explain why the corresponding torus knotgroup can be naturally considered as its braid group. In particular, this yields a new infinite familyof reflection-like groups admitting braid groups that are Garside groups. Moreover, we show that atoric reflection group has cyclic center by showing that the quotient by the center is isomorphic to thealternating subgroup of a Coxeter group of rank three. To this end we use the fact that the center of thealternating subgroup of an irreducible, infinite Coxeter group of rank at least three is trivial. Severalingredients of the proofs are purely Coxeter-theoretic, and might be of independent interes