On $p$–almost direct products and residual properties of pure braid groups of nonorientable surfaces

Abstract

We prove that the $n^{th}$ pure braid group of a nonorientable surface (closed or with boundary, but different from $ℝ{\mathbb{P}}^2$) 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.