On virtual singular braid groups

Abstract The virtual singular braid group arises as a natural common generalization of classical singular braid groups and virtual braid groups. In this paper, we study several algebraic properties of the virtual singular braid group VSG n \mathrm{VSG}_{n} . We introduce numerical invariants for virtual singular braids arising from exponent sums of words in VSG n \mathrm{VSG}_{n} , and describe explicitly the kernels of the associated homomorphisms onto abelian groups. We then determine all group homomorphisms, up to conjugation, from VSG n \mathrm{VSG}_{n} to the symmetric group S n S_{n} , and obtain corresponding semi-direct product decompositions. In the particular case n = 2 n=2 , we provide explicit presentations and algebraic descriptions of the kernels. Moreover, we show that certain relations are forbidden in VSG n \mathrm{VSG}_{n} , and we introduce and study natural quotients of the virtual singular braid group, including welded and unrestricted versions, for which analogous structural results are obtained.