Structural aspects of virtual twin groups

Study of certain isotopy classes of a finite collection of immersed circles without triple or higher intersections on closed oriented surfaces is considered as a planar analogue of virtual knot theory with the genus zero case corresponding to classical knot theory. Alexander and Markov theorems are known in this setting with the role of groups being played by a class of right-angled Coxeter groups called twin groups, denoted $T_n$, in the genus zero case. For the higher genus case, the role of groups is played by a new class of groups called virtual twin groups, denoted $VT_n$. A virtual twin group $VT_n$ contains the twin group $T_n$ and the pure virtual twin group $PVT_n$, an analogue of the pure braid group. The paper investigates in detail some structural aspects of these groups. We prove that the pure virtual twin group $PVT_n$ is an irreducible right-angled Artin group with trivial center and give it's precise presentation. We show that $PVT_n$ has a decomposition as an iterated semidirect product of infinite rank free groups. We also give a complete description of the automorphism group of $PVT_n$. As applications, we show that $VT_n$ is residually finite and $PVT_n$ has the $R_\infty$-property. Along the way, we also obtain a presentation of $VT_n'$ and a freeness result on $PVT_n'$.