Bounded generation of SL_2 over rings of S-integers with infinitely many units

Abstract

Let O be the ring of S-integers in a number field k. We prove that if the group of units O^* is infinite then every matrix in $\Gamma$ = SL_2(O) is a product of at most 9 elementary matrices. This completes a long line of research in this direction. As a consequence, we obtain that $\Gamma$ is boundedly generated as an abstract group.

Final version - to appear in `Algebra and Number Theory'

Cited by in corpus (10)

• Non-virtually abelian anisotropic linear groups are not boundedly generated
• Strong boundedness of simply connected split Chevalley groups defined over rings
• Explicit strong boundedness for higher rank symplectic groups
• Words have bounded width in $SL(n,\mathbb{Z})$
• On the bounded generation of arithmetic ${\mathrm{SL}}_2$
• Idempotent factorizations of singular $2\times 2$ matrices over quadratic integer rings
• Width of SL(n,O_S,I)
• Elementary bounded generation for ${\mathrm{SL}}_n$ for global function fields and $n\ge 3$
• Bounded generation for congruence subgroups of ${\mathrm{Sp}}_4(R)$
• On $\mathfrak{P}$-adic continued fractions with extraneous denominators: some explicit finiteness results