Square-section braid groups and Higman-Neumann-Neumann extensions
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911221987409920 |
|---|---|
| author | Alvarado-Garduño, Omar González, Jesús |
| author_facet | Alvarado-Garduño, Omar González, Jesús |
| contents | For positive integers $n$, $p$ and $q$ with $pq-n>0$, let $UC(n,p\times q)$ denote the configuration space of $n$ unlabelled hard unit squares in the rectangle $[0,p]\times[0,q]$, and let $B_n(p\times q)$ denote the corresponding fundamental group. It is known that, as $pq-n$ becomes large, $UC(n,p\times q)$ starts capturing homotopical properties of the classical configuration space of $n$ unlabelled pairwise-distinct points in the plane. At the start of this approximation process, $UC(pq-1,p\times q)$ is homotopy equivalent to a wedge of $(p-1)(q-1)$ circles, while the only other general families of spaces $UC(n,p\times q)$ known to be aspherical are $UC(n,p\times2)$ for $p\geq n$, and $UC(pq-2,p\times q)$. The fundamental groups of the former family are known to be responsible for the ``right-angled'' relations in Artin's classical braid groups. We prove that the fundamental groups of the latter family have a minimal presentation all whose relators are commutators. In particular, after explaining how $B_{2p-2}(p\times2)$ arises as the right-angled Artin group (RAAG) associated to a certain meta-edge, we show that $B_{3p-2}(p\times3)$ is a Higman-Neumann-Neumann extension of the RAAG associated to the corresponding meta-square. We provide a geometric interpretation of the latter fact in terms of Salvetti complexes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17707 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Square-section braid groups and Higman-Neumann-Neumann extensions Alvarado-Garduño, Omar González, Jesús Geometric Topology Algebraic Topology 05C25, 20F05, 20F36, 20F65, 55R80, 57M07 For positive integers $n$, $p$ and $q$ with $pq-n>0$, let $UC(n,p\times q)$ denote the configuration space of $n$ unlabelled hard unit squares in the rectangle $[0,p]\times[0,q]$, and let $B_n(p\times q)$ denote the corresponding fundamental group. It is known that, as $pq-n$ becomes large, $UC(n,p\times q)$ starts capturing homotopical properties of the classical configuration space of $n$ unlabelled pairwise-distinct points in the plane. At the start of this approximation process, $UC(pq-1,p\times q)$ is homotopy equivalent to a wedge of $(p-1)(q-1)$ circles, while the only other general families of spaces $UC(n,p\times q)$ known to be aspherical are $UC(n,p\times2)$ for $p\geq n$, and $UC(pq-2,p\times q)$. The fundamental groups of the former family are known to be responsible for the ``right-angled'' relations in Artin's classical braid groups. We prove that the fundamental groups of the latter family have a minimal presentation all whose relators are commutators. In particular, after explaining how $B_{2p-2}(p\times2)$ arises as the right-angled Artin group (RAAG) associated to a certain meta-edge, we show that $B_{3p-2}(p\times3)$ is a Higman-Neumann-Neumann extension of the RAAG associated to the corresponding meta-square. We provide a geometric interpretation of the latter fact in terms of Salvetti complexes. |
| title | Square-section braid groups and Higman-Neumann-Neumann extensions |
| topic | Geometric Topology Algebraic Topology 05C25, 20F05, 20F36, 20F65, 55R80, 57M07 |
| url | https://arxiv.org/abs/2510.17707 |