Bounded generation of relative subgroups in Chevalley groups

N. A. Vavilov

Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 6, Tome 523 (2023), pp. 7-18

Résumé

The problem of bounded elementary generation is now completely settled for all Chevalley groups of rank $\ge 2$ over arbitrary Dedekind rings $R$ of arithmetic type with the fraction field $K$, with uniform bounds. Namely, for every reduced irreducible root system $\Phi$ of rank $\ge 2$ there exists a uniform bound $L=L(\Phi)$ such that the simply connected Chevalley groups $\mathrm G(\Phi,R)$ have elementary width $\le L$ for all Dedekind rings of arithmetic type, [18]. It is natural to ask, whether similar result holds for the relative elementary groups $E(\Phi,R,I)$, where $I\unlhd R$. Mating the usual rewriting argument, already invoked in this context by Tavgen [28], with the universal localisation by Stepanov [25], we can give a very short proof that this is indeed the case. In other words, the width of $E(\Phi,R,I)$ in elementary conjugates $z_{\alpha}(\xi,\zeta)=x_{-\alpha}(\zeta)x_{\alpha}(\xi)x_{-\alpha}(-\zeta)$, where $\alpha\in\Phi$, $\xi\in I$, $\zeta\in R$, is indeed bounded by some constant $M=M(\Phi,R,I)$. However, the resulting bounds $M$ are not uniform, they depend on the pair $(R,I)$.