Congruence subgroups of braid groups and crystallographic quotients. Part II
Paolo Bellingeri, Celeste Damiani, Oscar Ocampo, Charalampos Stylianakis
TL;DR
The article advances the understanding of congruence subgroups of the braid group $B_3$ by analyzing the full central, derived, and lower central series quotients of level-$m$ subgroups $B_3[m]$ for odd primes $m$ and $m=4$. It establishes that $B_3/[B_3[m],B_3[m]]$ is crystallographic, and that $B_3/\Gamma_k(B_3[m])$ is almost-crystallographic for all $k\ge2$, with explicit holonomy groups and dimension formulas. Central to the approach are the symplectic Burau representations, the decomposition $B_3[m]\cong\mathbb{Z}\times F_M$ (or variants) and the induced actions on lower central series quotients, yielding detailed descriptions in terms of free groups and finite quotients like $\mathrm{Sp}_2(\mathbb{Z}/m\mathbb{Z})/Z(\mathrm{Sp}_2(\mathbb{Z}/m\mathbb{Z}))$. The paper also isolates a Bieberbach instance at $p=3$ with holonomy $A_4\cong\mathrm{PSp}_2(\mathbb{Z}/3\mathbb{Z})$, and provides an appendix on the finite group $\rho_4(B_3)$ to support the crystallographic conclusions. Overall, the results illuminate how congruence subgroups of $B_3$ naturally give rise to almost-crystallographic structures with explicit holonomy, enriching the link between braid groups, mapping class groups, and crystallographic geometry.
Abstract
Following previous work on congruence subgroups and crystallographic braid groups, we study the lower central series of congruence braid groups related to the braid group $B_3$, showing in particular that corresponding quotients are almost crystallographic.