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.