Embeddings of the braid groups of covering spaces, classification of the finite subgroups of the braid groups of the real projective plane, and linearity of braid groups of low-genus surfaces

Daciberg Lima Gonçalves (1) , John Guaschi (2)

1 IME - Instituto de Matemática e Estatística
2 LMNO - Laboratoire de Mathématiques Nicolas Oresme

Let M be a compact, connected surface, possibly with a finite set of points removed from its interior. Let d,n be positive integers, and let N 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}(N) of N, 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.

Mots clés: braid groups; covering spaces; finite subgroups; mapping class group; linearity

Domaines: Topologie géométrique [math.GT]; Théorie des groupes [math.GR]

Daciberg Lima Gonçalves, John Guaschi. Embeddings of the braid groups of covering spaces, classification of the finite subgroups of the braid groups of the real projective plane, and linearity of braid groups of low-genus surfaces. Journal of the London Mathematical Society, 2012, 85 (3), pp.855-868. ⟨10.1112/jlms/jdr071⟩. ⟨hal-00395533⟩