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

Let M be a compact, connected surface, possibly with a finite set of points removed from its interior. Let d, n P N, and let € M be a d-fold covering space of M. We show that the covering map induces an embedding of the nth braid group BnpMq of M in the dnth braid group Bdnp€ Mq of € M, and give several applications of this result. First, we classify the finite subgroups of the nth braid group of the real projective plane, from which we deduce an alternative proof of the classification of the finite subgroups of themapping class group of the n-punctured real projective plane due to Bujalance, Cirre and Gamboa. Secondly, using the linearity of Bn due to Bigelow and Krammer, we show that the braid groups of compact, connected surfaces of low genus are linear. 2000 AMS Subject Classification: 20F36 (primary); 20E28, 20F50, 57M10, 20H20 (secondary).