On normal subgroups of the braided Thompson groups

摘要

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