Quotients of braid groups by their congruence subgroups

The congruence subgroups of braid groups arise from a congruence condition on the integral reduced Burau representation $B_n \to \operatorname{GL}_{n-1}(\mathbb Z)$. We find the image of such congruence subgroups in $\operatorname{GL}_{n-1}(\mathbb Z)$ - an open problem posed by Dan Margalit in 2018. Additionally we characterize the quotients of braid groups by their congruence subgroups in terms of symplectic congruence subgroups

Congruence subgroups and crystallographic quotients of small Coxeter groups

Small Coxeter groups are precisely the ones for which the Tits representation is integral, which makes the study of their congruence subgroups relevant. The symmetric group $S_n$ has three natural extensions, namely, the braid group $B_n$, the twin group $T_n$ and the triplet group $L_n$. The latter two groups are small Coxeter groups, and play the role of braid groups under the Alexander-Markov correspondence for appropriate knot theories, with their pure subgroups admitting suitable hyperplane arrangements as Eilenberg-MacLane spaces. In this paper, we prove that the congruence subgroup property fails for infinite small Coxeter groups which are not virtually abelian. As an application, we deduce that the congruence subgroup property fails for both $T_n$ and $L_n$ when $n \ge 4$. We also determine subquotients of principal congruence subgroups of $T_n$, and identify the pure twin group $PT_n$ and the pure triplet group $PL_n$ with suitable principal congruence subgroups. Further, we investigate crystallographic quotients of these two families of small Coxeter groups, and prove that $T_n /PT_n^{'}$, $T_n/T_n^{''}$ and $L_n /PL_n^{'}$ are crystallographic groups. We also determine crystallographic dimensions of these groups and identify the holonomy representation of $T_n/T_n^{''}$.Comment: 25 pages, to appear in Forum Mathematicu

Parabolic subgroups of Garside groups II: ribbons

We introduce and investigate the ribbon groupoid associated with a Garside group. Under a technical hypothesis, we prove that this category is a Garside groupoid. We decompose this groupoid into a semi-direct product of two of its parabolic subgroupoids and provide a groupoid presentation. In order to established the latter result, we describe quasi-centralizers in Garside groups. All results hold in the particular case of Artin-Tits groups of spherical type