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Congruence subgroups of braid groups and crystallographic quotients. Part I

Paolo Bellingeri, Celeste Damiani, Oscar Ocampo, Charalampos Stylianakis

TL;DR

The paper addresses how congruence subgroups of braid groups, defined via the mod $m$ Burau/symplectic representation, interact with crystallographic quotients obtained from braid groups. It develops a general crystallographic criterion for extensions with free abelian kernels and applies it to quotients $B_n/[B_n[m],B_n[m]]$, linking their structure to holonomy given by the symplectic image $\rho_m(B_n)$. It then analyzes symmetric quotients arising from inclusions of congruence subgroups and proves isomorphisms in the odd-$m$ case, clarifying when Braids yield crystallographic groups and how these quotients decompose as finite-index subgroups with torsion-free abelian cores. Overall, the results illuminate the arithmetic-to-geometric bridge between congruence theory and crystallographic group theory in braid-group settings, and set the stage for further exploration of higher-dimensional holonomy and torsion properties.

Abstract

This paper is the first of a two part series devoted to describing relations between congruence and crystallographic braid groups. We recall and introduce some elements belonging to congruence braid groups and we establish some (iso)-morphisms between crystallographic braid groups and corresponding quotients of congruence braid groups.

Congruence subgroups of braid groups and crystallographic quotients. Part I

TL;DR

The paper addresses how congruence subgroups of braid groups, defined via the mod Burau/symplectic representation, interact with crystallographic quotients obtained from braid groups. It develops a general crystallographic criterion for extensions with free abelian kernels and applies it to quotients , linking their structure to holonomy given by the symplectic image . It then analyzes symmetric quotients arising from inclusions of congruence subgroups and proves isomorphisms in the odd- case, clarifying when Braids yield crystallographic groups and how these quotients decompose as finite-index subgroups with torsion-free abelian cores. Overall, the results illuminate the arithmetic-to-geometric bridge between congruence theory and crystallographic group theory in braid-group settings, and set the stage for further exploration of higher-dimensional holonomy and torsion properties.

Abstract

This paper is the first of a two part series devoted to describing relations between congruence and crystallographic braid groups. We recall and introduce some elements belonging to congruence braid groups and we establish some (iso)-morphisms between crystallographic braid groups and corresponding quotients of congruence braid groups.
Paper Structure (10 sections, 12 theorems, 26 equations, 4 figures)

This paper contains 10 sections, 12 theorems, 26 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Congruence subgroups
  3. Braid groups and examples
  4. Burau representation and symplectic structures
  5. Actions of half-twists on symplectic groups
  6. Crystallographic structures and congruence subgroups of the braid groups
  7. A general result on crystallographic groups
  8. Crystallographic structures and congruence subgroups of braid groups
  9. Symmetric quotients of congruence subgroups of braid groups
  10. Technical proofs

Key Result

Lemma 2.1

For $m\geq 2$ we have that $\rho_m(\sigma^m_i)=1$.

Figures (4)

  • Figure 1: Twist map acts on an Annulus.
  • Figure 2: The Dehn twist along the curve that surrounds the punctures $p_2,p_5$ is $A_{2,5}$.
  • Figure 3: Generators for $\mathrm{H}_1(\Sigma';\mathbb{Z})$ on the left, and $\mathrm{H}_1(\Sigma,Q;\mathbb{Z})$ on the right.
  • Figure 4: An example of a 2-fold cover of a marked disc. The simple closed curve $c_i$ in the genus 3 surface becomes the arc $a_i$ in the disc.

Theorems & Definitions (17)

  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Definition 3.1
  • Lemma 3.2: Goncalves-Guaschi-Ocampo:2017
  • Theorem 3.3
  • Theorem 3.4
  • Proposition 3.5
  • Theorem 3.6
  • Remark 3.7
  • ...and 7 more