Arithmetic Groups

From Andrei's talk, we have seen the significance of the notion of Bounded Generation in group theory. In this talk, we will show that if a linear group $\Gamma \subset \mathrm{GL}_n(K)$ over a field $K$ of characteristic zero is boundedly generated by semi-simple (diagonalizable) elements then it is virtually solvable. As a consequence, one obtains that infinite $S$-arithmetic subgroups of absolutely almost simple anisotropic algebraic groups over number fields are never boundedly generated. Our proof relies on Laurent's theorem from Diophantine geometry and properties of generic elements. This is joint work with Corvaja, Rapinchuk, and Zannier.