Almost-crystallographic groups as quotients of Artin braid groups (1805.11376v2)

Abstract: 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 $B_5/\Gamma_3 (P_5)$.