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"Super-approximation" in Absolutely Almost Simple Groups Over the Field of Rational Functions with Coefficients in a Finite Field


Let $p$ be a prime number greater than $5$, and let $q_0$ be a fixed power of $p$. Let $\bbf_{q_0}(t)$ be the field of rational functions with coefficients in the finite field $\bbf_{q_0}$ of order $q_0$. Let $\Omega\subset \GL_n(\bbf_{q_0}(t))$ be a finite symmetric set and let $\Gamma$ be the group generated by $\Omega$. Suppose the Zariski closure, $\bbg$, of $\Gamma$ is absolutely almost simple and simply connected, and that the ring generated by the set $\tr(\Ad\Gamma)$ is all of $\bbf_{q_0}[t,1/Q_0]$ where $Q_0$ is a common denominator of the entries of the matrices in $\Omega$. Then there exists a positive constant $\varepsilon>0$ depending only on $\bbg$ such that the set of Cayley graphs,

\{Cay(\pi_Q(\Gamma),\pi_Q(\Omega))\} forms a family of $\varepsilon$-expander graphs as $Q$ ranges through a suitable subset of the square free polynomials that are coprime to $Q_0$.

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