Superstrong approximation is a generalisation ofstrong approximation in algebraic groupsG, to providespectral gap results. The spectrum in question is that of theLaplacian matrix associated to a family of quotients of a discrete group Γ; and the gap is that between the first and second eigenvalues (normalisation so that the first eigenvalue corresponds to constant functions as eigenvectors). Here Γ is a subgroup of the rational points ofG, but need not be alattice: it may be a so-calledthin 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 forZariski dense subgroups Γ of thespecial linear group over the integers, and in more general classes of algebraic groupsG, is that the sequence ofCayley graphs for reductions Γ_p modulo prime numbersp, with respect to any fixed setS in Γ that is asymmetric set andgenerating set, is anexpander family.^[1]

In this context "strong approximation" is the statement thatS when reduced generates the full group of points ofG over the prime fields withp elements, whenp is large enough. It is equivalent to the Cayley graphs being connected (whenp is large enough), or that the locally constant functions on these graphs are constant, so that the eigenspace for the first eigenvalue is one-dimensional. Superstrong approximation therefore is a concrete quantitative improvement on these statements.

Property${\displaystyle (\tau )}$ is an analogue in discrete group theory ofKazhdan's property (T), and was introduced byAlexander Lubotzky.^[2] For a given family of normal subgroupsN of finite index in Γ, one equivalent formulation is that the Cayley graphs of the groups Γ/N, all with respect to a fixed symmetric set of generatorsS, form an expander family.^[3] Therefore superstrong approximation is a formulation of property${\displaystyle (\tau )}$, where the subgroupsN are the kernels of reduction modulo large enough primesp.

TheLubotzky–Weiss conjecture states (for special linear groups and reduction modulo primes) that an expansion result of this kind holds independent of the choice ofS. For applications, it is also relevant to have results where the modulus is not restricted to being a prime.^[4]

Results on superstrong approximation have been found using techniques onapproximate subgroups, andgrowth rate in finite simple groups.^[5]

• Breuillard, Emmanuel; Oh, Hee, eds. (2014).Thin Groups and Superstrong Approximation. Cambridge University Press.ISBN 978-1-107-03685-7.
• Matthews, C. R.; Vaserstein, L. N.; Weisfeiler, B. (1984). "Congruence properties of Zariski-dense subgroups. I.".Proc. London Math. Soc. Series 3.48 (3):514–532.doi:10.1112/plms/s3-48.3.514.MR 0735226.

• Algebraic groups
• Cayley graphs
• Spectral theory