Toric reflection groups

TL;DR

This paper introduces and classifies a new family of infinite complex reflection groups called toric reflection groups, derived from quotients of torus knot groups, and explores their algebraic and geometric properties.

Contribution

It defines and classifies toric reflection groups, linking them to braid groups, Coxeter groups, and Garside structures, expanding the understanding of reflection groups of rank two.

Findings

Includes all finite rank-two complex reflection groups with a single conjugacy class of hyperplanes.

Provides a classification of toric reflection groups and their relation to braid groups.

Shows that these groups have cyclic centers and relate to Coxeter groups of rank three.

Abstract

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.