Let M be a compact surface without boundary, and $n\ge 2$. We analyse the quotient group $B_n(M)/{\mathrm{\Gamma }}_2(P_n(M))$ of the surface braid group $B_n(M)$ by the commutator subgroup ${\mathrm{\Gamma }}_2(P_n(M))$ of the pure braid group $P_n(M)$. If M is different from the 2-sphere ${\mathbb{S}}^2$, we prove that $B_n(M)/{\mathrm{\Gamma }}_2(P_n(M))\cong P_n(M)/{\mathrm{\Gamma }}_2(P_n(M)){⋊}_{\phi }{S}_n$, and that $B_n(M)/{\mathrm{\Gamma }}_2(P_n(M))$ 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 $B_n(M)/{\mathrm{\Gamma }}_2(P_n(M))$. 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 $B_n(M)/{\mathrm{\Gamma }}_2(P_n(M))$ isomorphic either to $S_n$ or to certain Frobenius groups. We prove that crystallographic groups whose image by the projection $B_n(M)/{\mathrm{\Gamma }}_2(P_n(M))⟶S_n$ is a Frobenius group are not Bieberbach groups. Finally, we construct a family of Bieberbach subgroups ${\tilde{G}}_{n,g}$ of $B_n(M)/{\mathrm{\Gamma }}_2(P_n(M))$ 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 Kähler manifold that admits Anosov diffeomorphisms.

令M为无边界的紧致曲面，并且$ñ\ge 2$。我们分析商群${乙}_{ñ}（\mathrm{中号}）/{\mathrm{\Gamma }}_2（P_{ñ}（\mathrm{中号}））$ 编织层组 ${乙}_{ñ}（\mathrm{中号}）$ 按换向器子组 ${\mathrm{\Gamma }}_2（P_{ñ}（\mathrm{中号}））$ 纯编织族 $P_{ñ}（\mathrm{中号}）$。如果M与2球面不同${\mathbb{小号}}^2$，我们证明 ${乙}_{ñ}（\mathrm{中号}）/{\mathrm{\Gamma }}_2（P_{ñ}（\mathrm{中号}））\cong P_{ñ}（\mathrm{中号}）/{\mathrm{\Gamma }}_2（P_{ñ}（\mathrm{中号}））{⋊}_{\phi }{\mathrm{小号}}_{ñ}$， 然后 ${乙}_{ñ}（\mathrm{中号}）/{\mathrm{\Gamma }}_2（P_{ñ}（\mathrm{中号}））$当且仅当M是可取向的时，是晶体组。

如果M是可定向的，我们证明关于M的结构的许多结果${乙}_{ñ}（\mathrm{中号}）/{\mathrm{\Gamma }}_2（P_{ñ}（\mathrm{中号}））$。我们表征该组的有限阶元素，并确定这些元素的共轭类。我们还表明，存在一个有限子群的单个共轭类${乙}_{ñ}（\mathrm{中号}）/{\mathrm{\Gamma }}_2（P_{ñ}（\mathrm{中号}））$ 同构 ${\mathrm{小号}}_{ñ}$或某些Frobenius团体。我们证明了那些晶体群的图像由投影${乙}_{ñ}（\mathrm{中号}）/{\mathrm{\Gamma }}_2（P_{ñ}（\mathrm{中号}））⟶{\mathrm{小号}}_{ñ}$是Frobenius团体而不是Bieberbach团体。最后，我们构建了一个Bieberbach子族${\stackrel{〜}{G}}_{ñ，G}$ 的 ${乙}_{ñ}（\mathrm{中号}）/{\mathrm{\Gamma }}_2（P_{ñ}（\mathrm{中号}））$ 尺寸 $2ñG$并且其整齐性群是阶n的有限循环群，并且如果${\mathcal{X}}_{ñ，G}$ 是一个平面流形，其基本群是 ${\stackrel{〜}{G}}_{ñ，G}$，我们证明它是可定向的Kähler流形，它接受Anosov微分同构。