Almost-crystallographic groups as quotients of Artin braid groups

Abstract

Let $n,k\ge 3$. 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 $n\ge 5$, 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 $B\_5/\Gamma \_3(P\_5)$.