Crystallographic structures and braid theory

Let $M$ be a compact surface without boundary, and $n\geq 2$. 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 $B_n(M)/\Gamma_2(P_n(M)) \cong P_n(M)/\Gamma_2(P_n(M)) \rtimes_{\varphi} S_n$, 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 show a number of results regarding the structure of $B_n(M)/\Gamma_2(P_n(M))$. Finally, we construct a family of Bieberbach subgroups $\widetilde{G}_{n,g}$ 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 $\widetilde{G}_{n,g}$, we prove that it is an orientable Kähler manifold that admits Anosov diffeomorphisms. Joint work with Daciberg Lima Gonçalves, John Guaschi and Carolina de Miranda e Pereiro.