The aim of this talk will be to give an introduction to the surface braid groups and moreover to present both the splitting problem of surface braid groups and certain results about this problem concerning the mixed braid groups of the projective plane.

Surface braid groups are both a generalisation to any connected surface of the funda- mental group of a surface and of the braid groups of the plane, which are known as Artin braid groups and were defined by Artin in 1925. They were initially introduced by Zariski and later during the 1960’s, Fox gave an equivalent definition of surface braid groups from a topological point of view.

In the first part of the talk, we will define the surface braid groups from both a geometric and a topological point of view and we will present their close relation to the symmetric groups. Moreover, we will present an important family of surface braid groups; the so-called mixed braid groups. Finally, we will describe the splitting problem of surface braid groups, which we will see in detail in the second part of the talk.

In the second part of the talk, we will focus on the splitting problem, which, during the 1960’s, the period of the development of the theory of surface braid groups, was studied by many mathematicians; notably by Fadell, Neuwirth, Van Buskirk and Birman, and more recently by Gonçalves–Guaschi and Chen–Salter. In particular, we will focus on the case of the projective plane, where we will present its braid groups as well as certain results that we obtained concerning the splitting problem of its mixed braid groups.