An exploration of normalish subgroups of R. Thompson's groups F and T

In this short note, we show that R. Thompson's group F admits a normalish amenable subgroup, and that the standard copy of F in R. Thompson's group T is normalish in T. We further conjecture that if F is non-amenable, then T does not admit a normalish amenable subgroup, and therefore that the reduced C^* algebra of T is in fact simple in that case.