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

John Guaschi (1, 2, 3) , Daniel Juan-Pineda (4)

1 UNICAEN - Université de Caen Normandie
2 NU - Normandie Université
3 LMNO - Laboratoire de Mathématiques Nicolas Oresme
4 CCM - Centro de Ciencias Matemáticas, UNAM

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.

Mots clés: surface braid group; sphere braid group; projective plane braid group; configuration space; lower algebraic $K$-theory; conjugacy classes; virtually cyclic subgroups; Farrell-Jones conjecture; Nil groups

Domaines: Topologie géométrique [math.GT]; K-théorie et homologie [math.KT]; Théorie des groupes [math.GR]

John Guaschi, Daniel Juan-Pineda. A survey of surface braid groups and the lower algebraic K-theory of their group rings. L. Ji, A. Papadopoulos and S.-T. Yau. Handbook of Group Actions, Volume II, 32, International Press of Boston Inc., pp.23-76, 2015, Advanced Lectures in Mathematics, 9781571463012. ⟨hal-00794539⟩