2 problems

Finiteness conjecture. The group $V(G)$ and its commutator subgroup $[V(G),V(G)]$ are of type $\operatorname{F}_\infty$.

Let $m\in\mathbb{N}\cup\{\infty\}$, let $\psi$ be the character considered in the surrounding discussion, and let the ambient group be one of the Lodha--Moore groups $G$, $G_y$,…