Superstrong approximation

Here Γ is a subgroup of the rational points of G, but need not be a lattice: it may be a so-called thin group.The "gap" in question is a lower bound (absolute constant) for the difference of those eigenvalues.A consequence and equivalent of this property, potentially holding for Zariski dense subgroups Γ of the special linear group over the integers, and in more general classes of algebraic groups G, is that the sequence of Cayley graphs for reductions Γp modulo prime numbers p, with respect to any fixed set S in Γ that is a symmetric set and generating set, is an expander family.[1] In this context "strong approximation" is the statement that S when reduced generates the full group of points of G over the prime fields with p elements, when p is large enough.is an analogue in discrete group theory of Kazhdan's property (T), and was introduced by Alexander Lubotzky.
strong approximation in algebraic groupsspectral gapLaplacian matrixlatticethin groupZariski densespecial linear groupCayley graphssymmetric setgenerating setexpander familyKazhdan's property (T)Alexander Lubotzkygrowth rateLubotzky, AlexNotices of the American Mathematical Society