  Several finite complex reflection groups have a braid group which is
isomorphic to a torus knot group. The reflection group is obtained from the
torus knot group by declaring meridians to have order $k$ for some $k\geq 2$,
and meridians are mapped to reflections. We study all possible quotients of
torus knot groups obtained by requiring meridians to have finite order. Using
the theory of $J$-groups of Achar and Aubert, we show that these groups behave
like (in general infinite) complex reflection groups of rank two. The large
family of "toric reflection groups" which we obtain includes, among others, all
finite complex reflection groups of rank two with a single conjugacy class of
reflecting hyperplanes, as well as Coxeter's truncations of the $3$-strand
braid group. We classify these toric reflection groups and explain why the
corresponding torus knot group can be naturally considered as its braid group.
In particular, this yields a new infinite family of reflection-like groups
admitting a braid group which is a Garside group. Moreover, we show that a
toric reflection group has cyclic center by showing that the quotient by the
center is isomorphic to the alternating subgroup of a Coxeter group of rank
three. To this end we use the fact that the center of the alternating subgroup
of an irreducible, infinite Coxeter group of rank at least three is trivial.
Several ingredients of the proofs are purely Coxeter-theoretic, and might be of
independent interest.
