Una breve descripción de un importantísimo resultado de Boris Weisfeiler, matemático desparecido en Chile en 1985.

1. El gran teorema
de Weisfeiler

2. https://www.somachi-weisfeiler.com/

3. • In the eigthies, most of Weisfeiler's research in theoretical
mathematics was devoted to Lie groups and its subgroups.
• Lie groups are generalizations of groups of matrices.
• These are a fundamental object in mathematics; the work of
outstanding mathematicians is related to them (Jordan, Lie, Klein,
Poincaré, Cartan...; Margulis, Thurston, Tits, Gromov, Milnor, Ratner,
Bourgain, Tao, Mirzakhani, Venkatesh,...)
• Some of the most important open questions in Mathematics have at
least a reformulation in terms of Lie (sub)groups and their quotients.
On the mathematics of Boris Weisfeiler

4. Groups
- A group is the mathematical object that encodes the symme- tries of
a system (Galois, Cayley, Noether, ...).
- A geometric example:

5. (Groups of) Matrices
• Matrices can be multiplied and “inverted”.
• Warning ! In general, the product AB is not equal to BA
• An example of matrices that do commute:
cos
2𝜋
𝑛
−sin
2𝜋
𝑛
0
sin
2𝜋
𝑛
cos
2𝜋
𝑛
0
0 0 1

6. From Jordan to Weisfeiler
Camille Jordan proved that every finite group of matrices
n x n with complex entries is “almost” commutative:
- There exists a finite-index subgroup of matrices that do commute
- The index depends only on n (the dimension)
- But how large is the index ? (how small is the percentage ?)

7. Theorem
(Weisfeiler):
Roughly, the
index is at
most 𝑛 2𝑛

8. Effective/quantitative theorems in Mathematics
• We know from Euclides that there are infinitely many prime
numbers.
• The list of prime numbers that are known by today is finite; the
largest prime number so far detected is:

9. What about infinite groups ?
• Take a finite family S of invertible matrices and consider products,
inverses, products, inverses... and so on (one thus “generates”
a group).
• Assume that matrices have integer entries.
• Assume that the matrices are not “trapped” by an algebraic relation
except for det = 1 (Zarisky density).
Question: how large is the group G generated by S ?
One way to address this question is by reducing matrices (mod p).

10. Theorem
(Weisfeiler):
for large prime
numbers p, the
reduction (mod p)
of the matrices
cover the whole
group SL(3, Zp)

11. Super-strong approximation
• The group Gp= SL(3,Zp) comes with a system of generators Sp
• As such, it has a Cayley graph:

12. These are expanders !
Theorem (Sarnak-Xue, Gamburd, Bourgain-Gamburd, Helfgott, Breuillard-
Green-Tao, Pyber-Szabo, Varjú, Salehi-Golsefidy-Varjú):
The sequence of graphs (Gp, Sp) is an expander !
(a sequence of graphs with larger and larger number of nodes but sharing
similar quantitative connectivity properties)

13. Expanders
Given a subset X of a graph G, a node is in the boundary of X if it is not in X but is
connected by and edge to a node of X.
Given δ>0, we say that a graph G is
δ-connected if every subset X of G with no
more than half of the points of G satisfies
| boundary of X | > δ |X|
An expander is a sequence of graphs with
larger and larger number of nodes that are
δ-connected for the same value of δ

14. Building expanders
 The existence of expanders was shown by Paul Erdös.
 The first explicit construction is due to Margulis.
 The fact that the graphs involved in Weisfeiler's theorem form a
expander is a spectacular recent achievement.
 This is a subject of great research activity on these days.
 This shows how important Weisfeiler's theorem is: it opened the
path to a deep and prolific theory.
MANY THANKS