Powers of half-twists and congruence subgroups of braid groups

2025/11/11 by Paolo Bellingeri, Céleste Damiani, Bellingeri, Paolo +5
Mathematics · #20F05 #20F65 #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Primary 20F36 #Rings, Modules, and Algebras #Secondary 20H15

paper · pdf · doi:10.48550/arxiv.2511.08472

openalex publication_date 2025/11/11 · openalex created_date 2025/11/13 · openalex updated_date 2026/07/28

Abstract

In this work, we study the relationship between congruence subgroups B_n[m] and N_n(σ₁^m) the normal closure of σ₁^m, where σ₁ is the classical generator of B_n. We characterize the conditions under which N_n(σ₁^m) has finite index in B_n[m] and provide explicit generators for these finite quotients. For the cases where the index is infinite, we show that B_n[m]/N_n(σ₁^m) contains a free subgroup. Additionally, we compute the Abelianisation of Coxeter braid subgroups in the finite index cases and construct new finite quotients using commutators of congruence subgroups.

Citations

• Coxeter-type quotients of surface braid groups
• Generators for the level m congruence subgroups of braid groups
• Congruence subgroups of braid groups and crystallographic quotients. Part II
• Quotients of braid groups by their congruence subgroups
• The Cohomology of the Mod 4 Braid Group
• The normal closure of a power of a half-twist has infinite index in the\n mapping class group of a punctured sphere

Related

• Generators for the level m congruence subgroups of braid groups
• Abelian quotients for groups generated by cubed half-twists
• Quotients of braid groups by their congruence subgroups
• Congruence subgroups of braid groups
• On quotients of congruence subgroups of braid groups