Non-virtually abelian anisotropic linear groups are not boundedly generated

TL;DR

The paper proves that linear groups over characteristic zero fields, boundedly generated by semi-simple elements, are virtually solvable, impacting the understanding of bounded generation in algebraic groups and their subgroups.

Contribution

It establishes a new criterion linking bounded generation by semi-simple elements to virtual solvability in linear groups over characteristic zero fields.

Findings

Boundedly generated linear groups by semi-simple elements are virtually solvable.

Infinite S-arithmetic subgroups of anisotropic algebraic groups are not boundedly generated.

The proof uses Laurent's theorem and properties of generic elements.

Abstract

We prove 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.