# Superstrong approximation

**Superstrong approximation** is a generalisation of strong approximation in algebraic groups *G*, to provide "spectral gap" results. The spectrum in question is that of the Laplacian 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 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 Γ_{p} 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. It is equivalent to the Cayley graphs being connected (when *p* 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.

## Background

**Property (τ)** is an analogue in discrete group theory of Kazhdan's property (T), and was introduced by Alexander Lubotzky.^{[2]} For a given family of normal subgroups *N* 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 generators *S*, form an expander family.^{[3]} Therefore superstrong approximation is a formulation of property (τ), where the subgroups *N* are the kernels of reduction modulo large enough primes *p*.

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

## Proofs of superstrong approximation

Results on superstrong approximation have been found using techniques on approximate subgroups, and growth rate in finite simple groups.^{[5]}

## Notes

- ↑ (Breuillard Oh)
- ↑ http://www.ams.org/notices/200506/what-is.pdf
- ↑ Alexander Lubotzky (1 January 1994).
*Discrete Groups, Expanding Graphs and Invariant Measures*. Springer. p. 49. ISBN 978-3-7643-5075-8. https://books.google.com/books?id=aNURlzNuotEC&pg=PA49. - ↑ (Breuillard Oh)
- ↑ (Breuillard Oh)

## References

- Breuillard, Emmanuel; Oh, Hee, eds. (2014),
*Thin Groups and Superstrong Approximation*, Cambridge University Press, ISBN 978-1-107-03685-7, http://library.msri.org/books/Book61/index.html - 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

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