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 3 Let 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
- 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 .
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.