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

摘要

Let p be a prime number greater than 5, and let q0 be a fixed power of p. Let Fq0(t) be the field of rational functions with coefficients in the finite field Fq0 of order q0. Let O ⊂ GLn(Fq0(t)) be a finite symmetric set and let Gamma be the group generated by O. Suppose the Zariski closure, G, of Gamma is absolutely almost simple and simply connected, and that the ring generated by the set Tr(Adgamma) is all of Fq0 [t,1/Q0] where Q0 is a common denominator of the entries of the matrices in O. Then there exists a positive constant epsilon > 0 depending only on G such that the set of Cayley graphs, {Cay(pi Q(gamma), piQ(O))} forms a family of epsilon-expander graphs as Q ranges through a suitable subset of the square free polynomials that are coprime to Q0.