#### Volume 16, issue 1 (2016)

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

### Paolo Bellingeri and Sylvain Gervais

Algebraic & Geometric Topology 16 (2016) 547–568
##### Abstract

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

##### Keywords
pure braid groups, nonorientable surfaces, mod $p$ Torelli groups, residually $p$–finite, $p$–almost direct product, residually nilpotent, lower central series
##### Mathematical Subject Classification 2010
Primary: 20F14, 20F36, 57M05
Secondary: 20D15
##### Publication
Received: 27 January 2015
Revised: 7 July 2015
Accepted: 8 July 2015
Published: 23 February 2016
##### Authors
 Paolo Bellingeri Laboratoire de Mathématiques Nicolas Oresme Université de Caen CNRS UMR 6139 F-14000 Caen France http://www.math.unicaen.fr/~bellinge/ Sylvain Gervais Laboratoire de Mathématique Jean Leray Université de Nantes CNRS UMR 6629 2, rue de la Houssinière BP 92208 F-44322 Cedex 3 Nantes France http://www.math.sciences.univ-nantes.fr/~gervais/