Crystallographic groups and flat manifolds from surface braid groups

TL;DR

This paper investigates the structure of certain quotient groups derived from surface braid groups, characterizing their properties as crystallographic or Bieberbach groups, and constructs examples of flat manifolds with special geometric features.

Contribution

It provides a detailed analysis of the quotient group B_n(M)/Γ_2(P_n(M)), including conditions for crystallographic structure and the construction of Bieberbach subgroups with specific geometric properties.

Findings

B_n(M)/Γ_2(P_n(M)) is crystallographic iff M is orientable.

Finite-order elements and conjugacy classes are characterized.

Constructed Bieberbach subgroups lead to orientable Kähler manifolds with Anosov diffeomorphisms.

Abstract

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 $B_n(M)/\Gamma_2(P_n(M))$ 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 K\"ahler manifold that admits Anosov diffeomorphisms.