Say a finitely generated subgroup of (with fixed generating set ) has super-approximation with respect to a set of positive integers which are coprime to if the family of Cayley graphs is an expander. Here,
Example. When is the set of powers of a single prime , the inverse limite of finite groups is the closure of in , equals -adic integers.
Expansion is expressible in terms of convolution of measures. Let denote the uniform probability measure on given generating set . Let be a compact group containing . Let denote the averaging operator
Definition 1 Let denote the operator norm of on the orthogonal complement of constant functions in of the closure of ,
Then super-approximation is equivalent to
Example. When is the set of powers of a single prime , super-approximation is equivalent to
Follow from work of many people.
2.1. Arithmetic lattices
Theorem 2 Let be a semi-simple -group, then has super-approximation (if it is infinite).
Selberg showed this for . Burger-Sarnak showed that this property passes from lattices in one group to another. Jacquet-Langlands used these to handle real rank one groups, with the exception of unitary groups, solved by Clozel. Clozel-Ullmo did the -adic case. In higher rank, Kazhdan-Margulis.
2.2. From arithmetic to more general groups
Theorem 3 (Bourgain-Varju) Let be Zariski-dense in . Then has super-approximation.
This relies on the dynamics of on the torus, classification of invariant measures. The method is limited to archimedean fields.
Theorem 4 (Salehi Golsefidy-Varju) Let have Zariski-closure . Then has super-approximation with respect to -th powers of all square-free integers coprime to iff .
Infinite abelianization easily make spectral gap impossible. It is the converse which is hard.
Theorem 5 (Salehi Golsefidy) Let have Zariski-closure . Then has super-approximation with respect to all powers of all primes coprime to iff .
Theorem 6 (Salehi Golsefidy-Zhang) Let have Zariski-closures and . Assume that is the smallest normal subgroup of containing . Then Then has super-approximation if does.
This provides new examples, among subgroups of arithmetic lattices.
Strong approximation describes the closure of in compact groups. Combined with super-approximation, this leads to interesting results.
3.1. Affine sieve
Theorem 7 (Salehi Golsefidy-Sarnak) Let , with Zariski closure , . Let be a rational polynomial that does not vanish identically on . There exist integers and such that
is Zariski-dense in .
This is in the spirit of Dirichlet’s theorem on primes in arithmetic progressions: we produce elements with few prime factors in the set of values of some polynomial on a subgroup. A special case is Oh-Kantorovitch’s work on inverse radii of Appolonian circles.
3.2. Sieve in groups
Theorem 8 (Lubotzky-Meiri) A non virtually solvable finitely generated subgroup of ( of characteristic 0) is not covered by finitely many shifts of its powers .
3.3. Orbit-equivalence rigidity
Let be a dense subgroup in a -adic analytic semisimple Lie group . Assume is represented by matrics with algebraic entries in some basis. Let be a locally profinite group, and a dense countable subgroup. Then the actions of on and of on are measurably orbit equvalent, then there is an isomorphism between open subgroups of and which maps (the intersections of) to .
Question. Does this imply that and are isomorphic ?
3.4. Deformations of Galois representations
Ellenberg-Hall-Kowalski use super-approximation to give a new proof of a result of Cadoret-Tamagawa on abelian schemes over curves.