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

TL;DR

This paper proves that pure braid groups on nonorientable surfaces are residually 2-finite and residually nilpotent, introducing p-almost direct products to analyze their structure and properties.

Contribution

It introduces the concept of p-almost direct products and applies it to establish residual properties of pure braid groups on nonorientable surfaces.

Findings

Pure braid groups are residually 2-finite.

Pure braid groups are residually nilpotent.

Develops theory of p-almost direct products.

Abstract

We prove that the n th pure braid group of a nonorientable surface (closed or with boundary, but different from RP2) 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 prove therefore also some results on lower central series and augmentation ideals of p-almost direct products.