Congruence subgroups of small Artin and Coxeter groups

TL;DR

This paper investigates the properties of small Artin and Coxeter groups, focusing on their congruence subgroups, and extends classical concepts from braid groups to a broader algebraic context.

Contribution

It introduces the notion of small Artin groups via an integral representation and studies their congruence subgroup property, identifying principal congruence subgroups at small levels.

Findings

Identification of Coxeter and Artin groups with the congruence subgroup property

Characterization of principal congruence subgroups at small levels

Extension of classical congruence subgroup concepts to small Artin groups

Abstract

Small Coxeter groups are exactly those for which the Tits representation takes integral values, which makes the study of their congruence subgroups significant. In \cite{MR0938643}, Squier introduced a matrix representation of an Artin group defined over the ring $\mathbb Z[s^{\pm}, t^{\pm}]$ of Laurent polynomials in two variables. This representation simultaneously generalises the Tits representation of the associated Coxeter groups and the reduced Burau representation of braid groups. We define small Artin groups as those for which this representation becomes integral when evaluated at $s=1$ and $t=-1$. Consequently, the study of congruence subgroups of small Artin groups extends the classical notion of congruence subgroups of braid groups, which arise from the integral reduced Burau representation. In this paper, we examine Coxeter and Artin groups that possess the congruence subgroup property and identify several of their principal congruence subgroups at small levels.