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

Abstract

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.

17 pages

References in corpus (7)

• The mapping class group of a genus two surface is linear
• The lower central and derived series of the braid groups of the sphere and the punctured sphere
• On the linearity of certain mapping class groups
• The classification and the conjugacy classes of the finite subgroups of the sphere braid groups
• Classification of the virtually cyclic subgroups of the pure braid groups of the projective plane
• The braid groups of the projective plane and the Fadell-Neuwirth short exact sequence
• The structure of the group of conjugating automorphisms and the linear representation of the braid groups of some manifolds

Cited by in corpus (8)

• The classification of the virtually cyclic subgroups of the sphere braid groups
• A survey of surface braid groups and the lower algebraic K-theory of their group rings
• Minimal generating and normally generating sets for the braid and mapping class groups of the disc, the sphere and the projective plane
• Abelian and metabelian quotients of surface braid groups
• Lower central series, surface braid groups, surjections and permutations
• Topological Switching via Exceptional Point Pairs in an Optical Microcavity Laser
• The braid groups $B_{n,m}(\mathbb{R}{P}^2)$ and the splitting problem of the generalised Fadell-Neuwirth short exact sequence
• Mapping class groups and function spaces: a survey