Virtual braids are a combinatorial generalization of braids. We present abstract braids as equivalence classes of braid diagrams on a surface, joining two distinguished boundary components. They are identified up to isotopy, compatibility, stability and Reidemeister moves. We show that virtual braids are in a bijective correspondence with abstract braids. Finally we demonstrate that for any abstract braid, its representative of minimal genus is unique up to compatibility and Reidemeister moves. The genus of such a representative is thus an invariant for virtual braids. We also give a complete proof of the fact that there is a bijective correspondence between virtually equivalent virtual braid diagrams and braid-Gauss diagrams.

International audienceWe show a simple and easily implementable solution to the word problem for virtual braid groups

We show that the complex of parabolic subgroups associated to the Artin-Tits group of type B is hyperbolic.

In this paper, we discuss algebraic, combinatorial and topological properties of singular virtual braids. On the algebraic side, we state the relations between classical and singular virtual objects, in addition we discuss a Birman-like conjecture for the virtual case. On the topological and combinatorial side, we prove that there is a bijection between singular abstract braids, singular horizontal Gauss diagrams up to a certain equivalence relation, and singular virtual braids, in particular using singular horizontal Gauss diagrams we obtain a presentation of the singular pure virtual braid monoid.