On virtual singular braid groups

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>VSG</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> VSGₙ . We introduce numerical invariants for virtual singular braids arising from exponent sums of words in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>VSG</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> VSGₙ , and describe explicitly the kernels of the associated homomorphisms onto abelian groups. We then determine all group homomorphisms, up to conjugation, from <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>VSG</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> VSGₙ to the symmetric group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>S</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> Sₙ , and obtain corresponding semi-direct product decompositions. In the particular case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> n=2 , we provide explicit presentations and algebraic descriptions of the kernels. Moreover, we show that certain relations are forbidden in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>VSG</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> VSGₙ , 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.