Gonçalves, Daciberg Lima; Guaschi, John Surface braid groups and coverings. (English) Zbl 1282.20037 J. Lond. Math. Soc., II. Ser. 85, No. 3, 855-868 (2012). Summary: Let \(M\) be a compact, connected surface, possibly with a finite set of points removed from its interior. Let \(d,n\in\mathbb N\), and let \(\widetilde M\) be a \(d\)-fold covering space of \(M\). We show that the covering map induces an embedding of the \(n\)-th braid group \(B_n(M)\) of \(M\) in the \(dn\)-th braid group \(B_{dn}(\widetilde M)\) of \(\widetilde M\), and give several applications of this result. First, we classify the finite subgroups of the \(n\)-th braid group of the real projective plane, from which we deduce an alternative proof of the classification of the finite subgroups of the mapping class group of the \(n\)-punctured real projective plane due to Bujalance, Cirre and Gamboa. Secondly, using the linearity of \(B_n\) due to Bigelow and Krammer, we show that the braid groups of compact, connected surfaces of low genus are linear. Cited in 1 ReviewCited in 7 Documents MSC: 20F36 Braid groups; Artin groups 57M07 Topological methods in group theory 57M10 Covering spaces and low-dimensional topology Keywords:compact connected surfaces; covering spaces; covering maps; braid groups; finite subgroups; mapping class groups Citations:Zbl 1068.30034 × Cite Format Result Cite Review PDF Full Text: DOI arXiv HAL