Congruence subgroups of small Artin and Coxeter groups
Pravin Kumar
TL;DR
The paper analyzes the congruence subgroup property (CSP) for Coxeter and Artin groups using the generalised Burau/Tits representations, focusing on small (integral) cases. It develops a general CSP framework for abstract groups, proves CSP for infinite small affine Coxeter groups that are virtually abelian, and shows that Artin groups with non-affine components typically lack CSP with respect to their integral representations. It then identifies explicit level-2 and level-4 principal congruence subgroups for various small Artin groups, linking quotients to natural Weyl-group quotients and providing explicit lattice- and matrix-theoretic descriptions. These results clarify when CSP holds in this setting and give concrete descriptions of principal congruence subgroups in both level-2 and level-4 scenarios, with implications for the algebraic and homological structure of these 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.