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Towards super-approximation in positive characteristic

In this note we show that the family of Cayley graphs of a finitely generated subgroup of ${\rm GL}_{n_0}(\mathbb{F}_p(t))$ modulo some admissible square-free polynomials is a family of expanders under certain algebraic conditions. Here is a more precise formulation of our main result. For a positive integer $c_0$, we say a square-free polynomial is $c_0$-admissible if degree of irreducible factors of $f$ are distinct integers with prime factors at least $c_0$. Suppose $Ω$ is a finite symmetric subset of ${\rm GL}_{n_0}(\mathbb{F}_p(t))$, where $p$ is a prime more than $5$. Let $Γ$ be the group generated by $Ω$. Suppose the Zariski-closure of $Γ$ is connected, simply-connected, and absolutely almost simple; further assume that the field generated by the traces of ${\rm Ad}(Γ)$ is $\mathbb{F}_p(t)$. Then for some positive integer $c_0$ the family of Cayley graphs ${\rm Cay}(π_{f(x)}(Γ),π_{f(x)}(Ω))$ as $f$ ranges in the set of $c_0$-admissible polynomials is a family of expanders, where $π_{f(t)}$ is the quotient map for the congruence modulo $f(t)$.