Almost-crystallographic groups as quotients of Artin braid groups - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2017

Almost-crystallographic groups as quotients of Artin braid groups

Résumé

Let $n, k \geq 3$. In this paper, we analyse the quotient group $B_n/\Gamma_k(P_n)$ of the Artin braid group $B_n$ by the subgroup $\Gamma_k(P_n)$ belonging to the lower central series of the Artin pure braid group $P_n$. We prove that it is an almost-crystallographic group. We then focus more specifically on the case $k=3$. If $n \geq 5$, and if $\tau \in N$ is such that $gcd(\tau, 6) = 1$, we show that $B_n/\Gamma_3 (P_n)$ possesses torsion $\tau$ if and only if $S_n$ does, and we prove that there is a one-to-one correspondence between the conjugacy classes of elements of order $\tau$ in $B_n/\Gamma_3 (P_n)$ with those of elements of order $\tau$ in the symmetric group $S_n$. We also exhibit a presentation for the almost-crystallographic group $B_n/\Gamma_3 (P_n)$. Finally, we obtain some $4$-dimensional almost-Bieberbach subgroups of $B_3/\Gamma_3 (P_3)$, we explain how to obtain almost-Bieberbach subgroups of $B_4/\Gamma_3(P_4)$ and $B_3/\Gamma_4(P_3)$, and we exhibit explicit elements of order $5$ in $B_5/\Gamma_3 (P_5)$.
Fichier principal
Vignette du fichier
ggo2.pdf (319.13 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-01800719 , version 1 (27-05-2018)
hal-01800719 , version 2 (30-05-2018)

Identifiants

Citer

Daciberg Lima Gonçalves, John Guaschi, Oscar Ocampo. Almost-crystallographic groups as quotients of Artin braid groups. 2017. ⟨hal-01800719v1⟩
41 Consultations
71 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More