## 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 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 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 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 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, 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 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 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.