Non-virtually abelian anisotropic linear groups are not boundedly generated

2021/01/31 by Pietro Corvaja, Andrei Rapinchuk, Jinbo Ren +1 · 1 citation
Mathematics · #math.GR #math.NT #msc:11F06 #msc:11D72

paper · pdf · doi:10.1007/s00222-021-01064-y

Final version; to appear in Invent. Math

arxiv created 2021/07/08 · arxiv updated 2022/01/19

Abstract

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.

Cited by

• Bounded Generation for SL_n(Λ)