Non-virtually abelian anisotropic linear groups are not boundedly generated

Abstract

We prove that if a linear group Γ ⊂ GLn(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.