On \(p\)-almost direct products and residual properties of pure braid groups of nonorientable surfaces. (English) Zbl 1376.20035
Summary: We prove that the \(n\)th pure braid group of a nonorientable surface (closed or with boundary, but different from \(\mathbb{RP}^{2}\)) is residually 2-finite. Consequently, this group is residually nilpotent. The key ingredient in the closed case is the notion of \(p\)-almost direct product, which is a generalization of the notion of almost direct product. We also prove some results on lower central series and augmentation ideals of \(p\)-almost direct products.
MSC:
20F36 | Braid groups; Artin groups |
20E26 | Residual properties and generalizations; residually finite groups |
20F14 | Derived series, central series, and generalizations for groups |
57M07 | Topological methods in group theory |
20E22 | Extensions, wreath products, and other compositions of groups |