Non-virtually abelian anisotropic linear groups are not boundedly generated

• 成果类型: Article
• 署名作者: Corvaja, Pietro; Rapinchuk, Andrei S.; Ren, Jinbo; Zannier, Umberto M.
• 署名单位: University of Virginia; Scuola Normale Superiore di Pisa
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-021-01064-y
• 发表日期: 2022
• 页码: 1-26
• 关键词: cohomology; SUBGROUPS; GROWTH; sl2

摘要

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.