Bounded Generation of linear groups by diagonalisable elements

Venue:
Geb. 20.30, SR 2.058

Date:
15.02.2024

Speaker:
Segev Gonen Cohen

Time:
15:45 Uhr

Abstract: We say that an abstract group is boundedly generated if it can be written as a product of a finite number of cyclic subgroups. A classical result of Carter and Keller shows that for n > 2, the group SL(n,Z) is boundedly generated by unipotent matrices, this was used by Shalom to derive explicit Kazhdan constants for these groups.
A natural question is if we can have the opposite phenomenon, and boundedly generate linear groups using only semisimple (diagonalisable) elements?
The purpose of this talk is to present a recent negative answer to this question by Corvaja, Rapinchuk, Ren, and Zannier. We will start by studying bounded generation abstractly and sketching why it holds for SL(n,Z), before talking about the specific problem. The core of the proof relies on Laurent’s Theorem, a result from Diophantine Geometry; time permitting we will indicate a proof of this result using the relatively novel tools of ominimality.
No knowledge of any of these topics will be assumed.