Superstrong approximation is a generalisation of strong approximation in algebraic groups ''G'', to provide

spectral gap In mathematics, the spectral gap is the difference between the moduli of the two largest eigenvalues of a matrix or operator; alternately, it is sometimes taken as the smallest non-zero eigenvalue. Various theorems relate this difference to othe ...

results. The spectrum in question is that of the

Laplacian matrix In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph Laplac ...

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 Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...

: 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 In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the genera ...

over the integers, and in more general classes of algebraic groups ''G'', is that the sequence of

Cayley graph In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayle ...

s for reductions Γ_''p'' modulo prime numbers ''p'', with respect to any fixed set ''S'' in Γ that is a

symmetric set In mathematics, a nonempty subset of a group is said to be symmetric if it contains the inverses of all of its elements. Definition In set notation a subset S of a group G is called if whenever s \in S then the inverse of s also belongs to ...

and

generating set In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to ...

, is an expander family. 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) In mathematics, a locally compact topological group ''G'' has property (T) if the trivial representation is an isolated point in its unitary dual equipped with the Spectrum of a C*-algebra, Fell topology. Informally, this means that if ''G'' acts un ...

, and was introduced by

Alexander Lubotzky Alexander Lubotzky ( he, אלכסנדר לובוצקי; born 28 June 1956) is an Israeli mathematician and former politician who is currently a professor at the Weizmann Institute of Science and an adjunct professor at Yale University. He served ...

. 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. 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.

Proofs of superstrong approximation

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

Notes

References

* *{{citation, MR=0735226 , last1=Matthews, first1= C. R., last2= Vaserstein, first2= L. N., last3= Weisfeiler, first3= B. , title=Congruence properties of Zariski-dense subgroups. I. , journal=Proc. London Math. Soc. , series=Series 3, volume= 48 , year=1984, issue= 3, pages= 514–532, doi=10.1112/plms/s3-48.3.514 Algebraic groups Cayley graphs Spectral theory