On normal subgroups of the braided Thompson groups

Abstract

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