Nikolaus Conference 2021

Speaker: Thomas Gobet (Tours)

Title: Toric Reflection Groups


We introduce and study a three-parameter family of (in general infinite) reflection-like groups that includes, among other, finite complex reflection groups of rank two with a single conjugacy class of reflecting hyperplanes, as well as Coxeter's truncated braid groups on three strands. We give a classification of these groups, and show that they can naturally be associated with a torus knot group that behaves like their "braid group". We also show that they have a cyclic center, and that the quotient by their center is an alternating subgroup of a Coxeter group of rank three. This gives a new explanation of a phenomenon which was previously observed on a case-by-case basis for some finite complex reflection groups of rank two.

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