** 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 and a such that every finite simple group has a generating set of size 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 with the following significance. Let be a finite simple group of Lie type and of rank . With at most exceptions, pairs of elements of produce Cayley graphs that constitute an -expander.

We conjecture that there should be no exception provided more generators should be allowed (with ).

** 1.1. Superstrong approximation **

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

Theorem 3Let be a simply connected semisimple algebraic group over . If generates a Zariski dense subgroup of , then generates and the associated Cayley graphs form an expander, as prime tends to infinity.

We expect this to generalize from prime to integer prime to some .

See Salehi-Golsefidy’s talk on friday.

**2. The Bourgain-Gamburd method **

- Initial stage (): exponential non-concentration on proper subgroups.
- Middle stage (): -flattening lemma. Convolution quantitatively decreases measure.
- Final stage (): quasirandomness.

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

**3. Classification of approximate groups of **

Every approximate group is either

- bounded,
- very large.

Equivalently, subset grow under tripling (Product Theorem).

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

- ,
- is nilpotent,
- is contained in finitely many cosets of .

** 3.1. Consequences **

Diameter bound: partial results on Babai’s conjecture.

**4. What about alternating groups? **

There are non-trivial approximate subgroups. For instance (Pyber), inside , take , where is a long cycle and is generated by transpositions with disjoint supports.

**Question**. Is every approximate subgroup of close to an approximate subgroup of a proper subgroup?

**5. Proof of Product Theorem **

It relies on the determnation of subgroups of . 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 .

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