Non-virtually abelian anisotropic linear groups are not boundedly generated.
- Resource Type
- Article
- Authors
- Corvaja, Pietro; Rapinchuk, Andrei S.; Ren, Jinbo; Zannier, Umberto M.
- Source
- Inventiones Mathematicae. Jan2022, Vol. 227 Issue 1, p1-26. 26p.
- Subject
- *DIOPHANTINE approximation
*LAURENT series
*GEOMETRY
*INFINITE groups
- Language
- ISSN
- 0020-9910
We prove that if a linear group Γ ⊂ 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. [ABSTRACT FROM AUTHOR]