Notes of Alireza Salehi Golsefidy’s Oxford lecture 23-03-2017

Super-approximation

1. Definition

Say a finitely generated subgroup {\Gamma} of {Gl_n({\mathbb Q})} (with fixed generating set {\Omega}) has super-approximation with respect to a set {C} of positive integers which are coprime to {q_0} if the family of Cayley graphs {Cay(\pi_q(\Gamma),\pi_q(\Omega))} is an expander. Here,

\displaystyle  \begin{array}{rcl}  \pi_q:{\mathbb Z}[\frac{1}{q_0}]\rightarrow{\mathbb Z}[\frac{1}{q_0}]/q{\mathbb Z}[\frac{1}{q_0}]. \end{array}

Example. When {C} is the set of powers of a single prime {p}, the inverse limite of finite groups {\pi_{p^n}} is the closure of {\Gamma} in {Gl_n({\mathbb Z}_p)}, {{\mathbb Z}_p} equals {p}-adic integers.

Expansion is expressible in terms of convolution of measures. Let {\mu} denote the uniform probability measure on given generating set {\Omega}. Let {G} be a compact group containing {\Gamma}. Let {T} denote the averaging operator

Definition 1 Let {\lambda(\Omega;G)} denote the operator norm of {T} on the orthogonal complement of constant functions in {L^2} of the closure of {\Gamma},

\displaystyle  \begin{array}{rcl}  \lambda(\Omega;G)=\|T\|_{|L^2_0(\bar\Gamma}\| \end{array}

Then super-approximation is equivalent to

\displaystyle  \begin{array}{rcl}  \sup_{q\in C}\lambda(\Omega,Gl_n({\mathbb Z}/q{\mathbb Z}))<1. \end{array}

Example. When {C} is the set of powers of a single prime {p}, super-approximation is equivalent to

\displaystyle  \begin{array}{rcl}  \lambda(\Omega,Gl_n({\mathbb Z}_p))<1. \end{array}

2. Results

Follow from work of many people.

2.1. Arithmetic lattices

Theorem 2 Let {G} be a semi-simple {{\mathbb Q}}-group, then {\Gamma=G({\mathbb Z}[\frac{1}{q_0}])} has super-approximation (if it is infinite).

Selberg showed this for {Sl_2}. 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 {p}-adic case. In higher rank, Kazhdan-Margulis.

2.2. From arithmetic to more general groups

Theorem 3 (Bourgain-Varju) Let {\Gamma<SL_n({\mathbb Z})} be Zariski-dense in {Sl_n}. Then {\Gamma} has super-approximation.

This relies on the dynamics of {Sl_n({\mathbb Z})} on the torus, classification of invariant measures. The method is limited to archimedean fields.

Theorem 4 (Salehi Golsefidy-Varju) Let {\Gamma<G({\mathbb R})} have Zariski-closure {G^0}. Then {\Gamma} has super-approximation with respect to {N_0}-th powers of all square-free integers coprime to {q_0} iff {G^0=[G^0,G^0]}.

Infinite abelianization easily make spectral gap impossible. It is the converse which is hard.

Theorem 5 (Salehi Golsefidy) Let {\Gamma<G({\mathbb R})} have Zariski-closure {G^0}. Then {\Gamma} has super-approximation with respect to all powers of all primes coprime to {q_0} iff {G^0=[G^0,G^0]}.

Theorem 6 (Salehi Golsefidy-Zhang) Let {\Lambda<\Gamma<Gl_n({\mathbb Z}[\frac{1}{q_0}])} have Zariski-closures {H} and {G}. Assume that {G^0} is the smallest normal subgroup of {G^0} containing {H^0}. Then Then {\Gamma} has super-approximation if {\Lambda} does.

This provides new examples, among subgroups of arithmetic lattices.

3. Applications

Strong approximation describes the closure of {\Gamma} in compact groups. Combined with super-approximation, this leads to interesting results.

3.1. Affine sieve

Theorem 7 (Salehi Golsefidy-Sarnak) Let {\Gamma<Gl_n({\mathbb Z}[\frac{1}{q_0}])}, with Zariski closure {G}, {G^0=[G^0,G^0]}. Let {f} be a rational polynomial that does not vanish identically on {G^0}. There exist integers {r} and {q'_0} such that

\displaystyle  \begin{array}{rcl}  \Gamma_{r,q'_0}(f)=\{\gamma\in\Gamma\,;\,f(\gamma)=p_1\cdots p_{r'},\,r'\leq r,\,p_i\textrm{ primes in }{\mathbb Z}[\frac{1}{q_0}]\} \end{array}

is Zariski-dense in {G}.

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 {\Gamma} of {Gl_n(F)} ({F} of characteristic 0) is not covered by finitely many shifts of its powers {\bigcup_{m\geq 2}\Gamma^m}.

3.3. Orbit-equivalence rigidity

Let {\Gamma<G} be a dense subgroup in a {p}-adic analytic semisimple Lie group {G}. Assume {Ad(\Gamma)} is represented by matrics with algebraic entries in some basis. Let {H} be a locally profinite group, and {\Lambda<H} a dense countable subgroup. Then the actions of {\Gamma} on {G} and of {\Lambda} on {H} are measurably orbit equvalent, then there is an isomorphism between open subgroups of {G} and {H} which maps (the intersections of) {\Gamma} to {\Lambda}.

Question. Does this imply that {\Gamma} and {\Lambda} 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.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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