Pré-soutenance : The splitting problem for braid groups of the projective plane and a remarkable quotient of welded braid groups

This talk will be divided into two parts.

 The first part concerns surface braid groups and in particular the splitting problem of thegeneralised Fadell–Neuwirth short exact sequence of braid groups of the projective plane.The splitting of the Fadell–Neuwirth short exact sequences, which we will explicitly define,was a central question, during the foundation and the development of the theory of braid
groups during the 1960’s, studied by Fadell , Neuwirth, Van Buskirk and Birman, among others.The aim of this talk is to provide necessary and sufficients conditions under which the generalised Fadell–Neuwirth short
exact sequence of braid groups of the projective plane splits. The second part of this talk deals with welded and unrestricted virtual braid groups. The welded braid groups are a
3-dimensional analogue of the Artin braid groups and the unresticted virtual ones a quotient of the welded braid groups.The aim will be to introduce these groups, to study their structure as well as their torsion elements.That includes; we will characterise all possible finite images of  the unrestricted virtual braid group, and also we willdetermine the automorphism group of an important subgroup of it.In addition, we will give a complete characterisation of the torsion elements of the unrestricted virtual braid group andwe will prove that it contains a crystallographic group as a subgroup.