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Register a new userAlmost-crystallographic groups as quotients of Artin braid groups
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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 $B_5/Gamma_3 (P_5)$.
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Generalising previous results on classical braid groups by Artin and Lin, we determine the values of m, n $in$ N for which there exists a surjection between the n-and m-string braid groups of an orientable surface without boundary. This result is essentially based on specific properties of their lower central series, and the proof is completely combinatorial. We provide similar but partial results in the case of orientable surfaces with boundary components and of non-orientable surfaces without boundary. We give also several results about the classification of different representations of surface braid groups in symmetric groups.
We consider exact sequences and lower central series of surface braid groups and we explain how they can prove to be useful for obtaining representations for surface braid groups. In particular, using a completely algebraic framework, we describe the notion of extension of a representation introduced and studied recently by An and Ko and independently by Blanchet.
An infinitary version of braid groups has been considered as a direct limit of n-braid groups. However, we can imagine more complicated braids with infinitely many strings. We invetisgate basic properties especially when the number of strings is countable.
Let $M$ be a compact surface without boundary, and $ngeq 2$. We analyse the quotient group $B_n(M)/Gamma_2(P_n(M))$ of the surface braid group $B_{n}(M)$ by the commutator subgroup $Gamma_2(P_n(M))$ of the pure braid group $P_{n}(M)$. If $M$ is different from the $2$-sphere $mathbb{S}^2$, we prove that $B_n(M)/Gamma_2(P_n(M))$ is isomorphic rho $P_n(M)/Gamma_2(P_n(M)) rtimes_{varphi} S_n$, and that $B_n(M)/Gamma_2(P_n(M))$ is a crystallographic group if and only if $M$ is orientable. If $M$ is orientable, we prove a number of results regarding the structure of $B_n(M)/Gamma_2(P_n(M))$. We characterise the finite-order elements of this group, and we determine the conjugacy classes of these elements. We also show that there is a single conjugacy class of finite subgroups of $B_n(M)/Gamma_2(P_n(M))$ isomorphic either to $S_n$ or to certain Frobenius groups. We prove that crystallographic groups whose image by the projection $B_n(M)/Gamma_2(P_n(M))to S_n$ is a Frobenius group are not Bieberbach groups. Finally, we construct a family of Bieberbach subgroups $tilde{G}_{n,g}$ of $B_n(M)/Gamma_2(P_n(M))$ of dimension $2ng$ and whose holonomy group is the finite cyclic group of order $n$, and if $mathcal{X}_{n,g}$ is a flat manifold whose fundamental group is $tilde{G}_{n,g}$, we prove that it is an orientable Kahler manifold that admits Anosov diffeomorphisms.
Let M be a compact surface, either orientable or non-orientable. We study the lower central and derived series of the braid and pure braid groups of M in order to determine the values of n for which B_n(M) and P_n(M) are residually nilpotent or residually soluble. First, we solve this problem for the case where M is the 2-torus. We then give a general description of these series for an arbitrary semi-direct product that allows us to calculate explicitly the lower central series of P_2(K), where K is the Klein bottle, and to give an estimate for the derived series of P_n(K). Finally, if M is a non-orientable compact surface without boundary, we determine the values of n for which B_n(M) is residually nilpotent or residually soluble in the cases that were not already known in the literature.
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