Crystallographic groups and flat manifolds from surface braid groups

Daciberg Lima Gonçalves; John Guaschi; Oscar Ocampo; Carolina de Miranda E Pereiro

doi:10.1016/j.topol.2020.107560

paper

Crystallographic groups and flat manifolds from surface braid groups

arXiv:2107.03683 · doi:10.1016/j.topol.2020.107560

Abstract

Let $M$ be a compact surface without boundary, and $n \geq 2$ . We analyse the quotient group $B_{n} (M) / Γ_{2} (P_{n} (M))$ of the surface braid group $B_{n} (M)$ by the commutator subgroup $Γ_{2} (P_{n} (M))$ of the pure braid group $P_{n} (M)$ . If $M$ is different from the $2$ -sphere $S^{2}$ , we prove that $B_{n} (M) / Γ_{2} (P_{n} (M))$ is isomorphic rho $P_{n} (M) / Γ_{2} (P_{n} (M)) ⋊_{φ} S_{n}$ , and that $B_{n} (M) / Γ_{2} (P_{n} (M))$ 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 $B_{n} (M) / Γ_{2} (P_{n} (M))$ . 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 $B_{n} (M) / Γ_{2} (P_{n} (M))$ isomorphic either to $S_{n}$ or to certain Frobenius groups. We prove that crystallographic groups whose image by the projection $B_{n} (M) / Γ_{2} (P_{n} (M)) \to S_{n}$ is a Frobenius group are not Bieberbach groups. Finally, we construct a family of Bieberbach subgroups $G ~_{n, g}$ of $B_{n} (M) / Γ_{2} (P_{n} (M))$ of dimension $2 n g$ and whose holonomy group is the finite cyclic group of order $n$ , and if $X_{n, g}$ is a flat manifold whose fundamental group is $G ~_{n, g}$ , we prove that it is an orientable Kähler manifold that admits Anosov diffeomorphisms.