3 problems

Let $F$ be a field, let $G_1$ be an absolutely almost simple algebraic $F$-group, and let $\Gamma_1\subset G_1(F)$ be a finitely generated Zariski-dense subgroup with trace field…

Let $G$ be an absolutely almost simple algebraic group over a field $K$ of good characteristic, and let $\Gamma\subset G(K)$ be a finitely generated Zariski-dense subgroup whose fi…

Let $F$ be the field occurring in the definition of weak commensurability. Let $G_1$ and $G_2$ be absolutely simple algebraic $F$-groups of adjoint type, and let $Gamma_1$ be a fin…