## Notes of Emmanuel Breuillard’s second Oxford lecture 21-03-2017

Approximate groups, II

1. Expansion in Cayley graphs

Solvable groups cannot lead to expanders. On the other hand, simple groups do.

Theorem 1 (Kassabov-Lubotsky-Nikolov) There is a ${k>0}$ and a ${\epsilon>0}$ such that every finite simple group has a generating set of size ${\leq k}$ that makes the collection of corresponding Cayley graphs an expander.

In fact, Suzuki groups were excluded from the theorem, until Breuillard-Green-Tao filled in this case.

Theorem 2 (Breuillard-Green-Guralnick-Tao) There exists ${\epsilon=\epsilon(r)}$ with the following significance. Let ${G}$ be a finite simple group of Lie type and of rank ${r}$. With at most ${|G|^{2-\epsilon}}$ exceptions, pairs of elements of ${G}$ produce Cayley graphs that constitute an ${\epsilon}$-expander.

We conjecture that there should be no exception provided more generators should be allowed (with ${\epsilon=\epsilon(r,k)}$).

1.1. Superstrong approximation

The following improves on the classical Weisfeiler-Matthew-Vaserstein theorem. It results from the efforts of many people.

Theorem 3 Let ${G}$ be a simply connected semisimple algebraic group over ${{\mathbb Q}}$. If ${S}$ generates a Zariski dense subgroup of ${G({\mathbb Q})}$, then ${S\mod p}$ generates ${G({\mathbb Z}/p{\mathbb Z})}$ and the associated Cayley graphs form an expander, as prime ${p}$ tends to infinity.

We expect this to generalize from ${p}$ prime to ${n}$ integer prime to some ${n_0}$.

See Salehi-Golsefidy’s talk on friday.

2. The Bourgain-Gamburd method

1. Initial stage (${n\leq c\log|G|}$): exponential non-concentration on proper subgroups.
2. Middle stage (${c\log|G|\leq n\leq C\log|G|}$): ${\ell^2}$-flattening lemma. Convolution quantitatively decreases measure.
3. Final stage (${n\geq C\log|G|}$): quasirandomness.

The middle stage involves approximate groups. It is similar to Balog-Gowers-Szemeredi’s theorem.

3. Classification of approximate groups of ${G(q)}$

Every approximate group is either

1. bounded,
2. very large.

Equivalently, subset grow under tripling (Product Theorem).

Conjecturally, the theorem should extend to other linear groups in the following form: if ${A}$ generates an approximate subgroup, then there exist subgroups ${N}$ and ${H}$ normalized by ${A}$, such that

1. ${N\subset A\cdots A}$,
2. ${H/N}$ is nilpotent,
3. ${A}$ is contained in finitely many cosets of ${H}$.

3.1. Consequences

Diameter bound: partial results on Babai’s conjecture.

There are non-trivial approximate subgroups. For instance (Pyber), inside ${\mathfrak{S}_n}$, take ${A=H\cup \{\sigma^{\pm 2}\}}$, where ${\sigma}$ is a long cycle and ${H}$ is generated by ${n/2}$ transpositions with disjoint supports.

Question. Is every approximate subgroup of ${\mathfrak{S}_n}$ close to an approximate subgroup of a proper subgroup?

5. Proof of Product Theorem

It relies on the determnation of subgroups of ${G(q)}$. Done by Weisfeiler, but it is a different argument, due to Larsen-Pink, which helps us. They prove a non-concentration estimate, in terms of size of intersection of subgroup with algebraic subvarieties of ${Gl_n(q)}$.

Hrushovski found an approximate group version of Larsen-Pink’s theorem.