We prove that if a linear group Gamma subset of 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.
Non-virtually abelian anisotropic linear groups are not boundedly generated
Corvaja P.;
2022-01-01
Abstract
We prove that if a linear group Gamma subset of 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.File in questo prodotto:
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