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 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.