On normal subgroups of the braided Thompson groups

Abstract

We inspect the normal subgroup structure of the braided Thompson groups Vbr and Fbr. We prove that every proper normal subgroup of Vbr lies in the kernel of the natural quotient Vbr \onto V, and we exhibit some families of interesting such normal subgroups. For Fbr, we prove that for any normal subgroup N of Fbr, either N is contained in the kernel of Fbr \onto F, or else N contains [Fbr,Fbr]. We also compute the Bieri-Neumann-Strebel invariant Sigma^1(Fbr), which is a useful tool for understanding normal subgroups containing the commutator subgroup.