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.