Towards super-approximation in positive characteristic

In this note we show that the family of Cayley graphs of a finitely generated subgroup of ${\mathrm{G}\mathrm{L}}_{n_0}({\mathbb{F}}_p(t))$ 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 $\mathrm{\Omega }$ is a finite symmetric subset of ${\rm GL}{n_0}($\mathbb{F}$p(t))$,where$p$isaprimemorethan$5$\mathrm{.}Let$\Gamma$bethegroupgeneratedby$\Omega$\mathrm{.}SupposetheZariski-closureof$\Gamma$ is connected, simply-connected, and absolutely almost simple; further assume that the field generated by the traces of $\mathrm{A}\mathrm{d}(\mathrm{\Gamma })$ is ${\mathbb{F}}_p(t)$. Then for some positive integer $c_0$ the family of Cayley graphs ${\rm Cay}(\pi{f(x)}(\Gamma),\pi{f(x)}(\Omega))$as$f$rangesinthesetof$c_0$-admissiblepolynomialsisafamilyofexpanders,where$\pi_{f(t)}$ is the quotient map for the congruence modulo $f(t)$.