Superstrong approximation

Superstrong approximation is a generalisation of strong approximation in algebraic groups to provide spectral gap results. A consequence of this property, 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]^[2]

Background

Property (τ) is an analogue in discrete group theory of Kazhdan's property (T), and was introduced by Alexander Lubotzky.^[3] 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.^[4] 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.^[5]

Proofs of superstrong approximation

Results on superstrong approximation have been found using techniques on approximate subgroups, and growth rate in finite simple groups.^[6]

Notes

^ Hee Oh; Emmanuel Breuillard (17 February 2014). Thin Groups and Superstrong Approximation. Cambridge University Press. p. x. ISBN 978-1-107-03685-7.
^ Hee Oh; Emmanuel Breuillard (17 February 2014). Thin Groups and Superstrong Approximation. Cambridge University Press. pp. 343–. ISBN 978-1-107-03685-7.
^ 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.
^ Hee Oh; Emmanuel Breuillard (17 February 2014). Thin Groups and Superstrong Approximation. Cambridge University Press. pp. 3–4. ISBN 978-1-107-03685-7.
^ Hee Oh; Emmanuel Breuillard (17 February 2014). Thin Groups and Superstrong Approximation. Cambridge University Press. p. xi. ISBN 978-1-107-03685-7.

References

Oh, Hee; Breuillard, Emmanuel, eds. (2014), Thin Groups and Superstrong Approximation, Cambridge University Press, ISBN 978-1-107-03685-7

[1] Hee Oh; Emmanuel Breuillard (17 February 2014). Thin Groups and Superstrong Approximation. Cambridge University Press. p. x. ISBN 978-1-107-03685-7.

[2] Hee Oh; Emmanuel Breuillard (17 February 2014). Thin Groups and Superstrong Approximation. Cambridge University Press. pp. 343–. ISBN 978-1-107-03685-7.

[3] ttp://www.ams.org/notices/200506/what-is.pdf

[Lubotzky1994-4] Alexander Lubotzky (1 January 1994). Discrete Groups, Expanding Graphs and Invariant Measures. Springer. p. 49. ISBN 978-3-7643-5075-8.

[OhBreuillard2014-5] Hee Oh; Emmanuel Breuillard (17 February 2014). Thin Groups and Superstrong Approximation. Cambridge University Press. pp. 3–4. ISBN 978-1-107-03685-7.

[6] Hee Oh; Emmanuel Breuillard (17 February 2014). Thin Groups and Superstrong Approximation. Cambridge University Press. p. xi. ISBN 978-1-107-03685-7.

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