Powers of half-twists and congruence subgroups of braid groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918196792000512 |
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| author | Bellingeri, Paolo Damiani, Celeste Ocampo, Oscar Stylianakis, Charalampos |
| author_facet | Bellingeri, Paolo Damiani, Celeste Ocampo, Oscar Stylianakis, Charalampos |
| contents | In this work, we study the relationship between congruence subgroups $B_n[m]$ and $\mathcal{N}_n(σ_1^m)$ the normal closure of $σ_1^m$, where $σ_1$ is the classical generator of $B_n$. We characterize the conditions under which $\mathcal{N}_n(σ_1^m)$ 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 $B_n[m]/\mathcal{N}_n(σ_1^m)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_08472 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Powers of half-twists and congruence subgroups of braid groups Bellingeri, Paolo Damiani, Celeste Ocampo, Oscar Stylianakis, Charalampos Group Theory Geometric Topology Primary 20F36, Secondary 20H15, 20F65, 20F05 In this work, we study the relationship between congruence subgroups $B_n[m]$ and $\mathcal{N}_n(σ_1^m)$ the normal closure of $σ_1^m$, where $σ_1$ is the classical generator of $B_n$. We characterize the conditions under which $\mathcal{N}_n(σ_1^m)$ 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 $B_n[m]/\mathcal{N}_n(σ_1^m)$ 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. |
| title | Powers of half-twists and congruence subgroups of braid groups |
| topic | Group Theory Geometric Topology Primary 20F36, Secondary 20H15, 20F65, 20F05 |
| url | https://arxiv.org/abs/2511.08472 |