A survey of surface braid groups and the lower algebraic K-theory of their group rings

Abstract

We give a survey of the theory of surface braid groups and the lower algebraic K-theory of their group rings. We recall several definitions and describe various properties of surface braid groups, such as the existence of torsion, orderability, linearity, and their relation both with mapping class groups and with the homotopy groups of the 2-sphere. The braid groups of the 2-sphere and the real projective plane are of particular interest because they possess elements of finite order, and we discuss in detail their torsion and the classification of their finite and virtually cyclic subgroups. Finally, we outline the methods used to study the lower algebraic K-theory of the group rings of surface braid groups, highlighting recent results concerning the braid groups of the 2-sphere and the real projective plane.

47 pages, 5 figures

References in corpus (12)

• The K-theoretic Farrell-Jones Conjecture for hyperbolic groups
• Braid Groups are Linear
• Braid structures in knot complements, handlebodies and 3-manifolds
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• On presentation of Surface Braid Groups
• Ordering pure braid groups on closed surfaces
• The lower central and derived series of the braid groups of the projective plane
• Exact sequences, lower central series and representations of surface braid groups
• K- and L-theory of group rings over GL_n(Z)

Cited by in corpus (5)

• On p-almost direct products and residual properties of pure braid groups of nonorientable surfaces
• The braid group for a quiver with superpotential
• Classifying spaces for the family of virtually cyclic subgroups of braid groups
• Polynomial braid combing
• The braid groups $B_{n,m}(\mathbb{R}{P}^2)$ and the splitting problem of the generalised Fadell-Neuwirth short exact sequence