Toward super‐approximation in positive characteristic

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