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arXiv · 2101.09386

Non-virtually abelian anisotropic linear groups are not boundedly generated

Pietro Corvaja·Andrei Rapinchuk·Jinbo Ren·Umberto Zannier
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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.

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Pietro Corvaja, Andrei Rapinchuk, Jinbo Ren, Umberto Zannier. 2021-01-23. Non-virtually abelian anisotropic linear groups are not boundedly generated. https://doi.org/10.1007/s00222-021-01064-y

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