Corvaja P, Rapinchuk A, Ren J, Zannier U. Non-virtually abelian anisotropic linear groups are not boundedly generated. Invent. math. 2022;227:1–26.

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## 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.

*Last updated on 09/25/2023*