Superstrong approximation
Superstrong approximation is a generalisation of
A consequence and equivalent of this property, potentially holding for
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 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 , 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 2014, pages x, 343)
- ^ Lubotzky, Alex (2005). "What is ... property ?" (PDF). MR 2147485.
- ISBN 978-3-7643-5075-8.
- ^ (Breuillard & Oh 2014, pages 3-4)
- ^ (Breuillard & Oh 2014, page xi)
References
- 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. MR 0735226.