Bounded generation of Steinberg groups over Dedekind rings of arithmetic type

TL;DR

This paper proves bounded elementary generation of Steinberg groups over Dedekind rings of arithmetic type, extending results to various root systems and polynomial rings, with uniform bounds based on Chevalley groups.

Contribution

It establishes bounded elementary generation for Steinberg groups over Dedekind rings of arithmetic type and polynomial rings, providing uniform bounds and extending previous results.

Findings

Bounded elementary generation for Steinberg groups over Dedekind rings of arithmetic type.

Bounded generation of Steinberg groups over polynomial rings $ ext{F}_q[t,t^{-1}]$ and $ ext{F}_q[t]$.

Uniform bounds derived from Chevalley groups.

Abstract

The main result of the present paper is bounded elementary generation of the Steinberg groups $\mathrm{St}(\Phi,R)$ for simply laced root systems $\Phi$ of rank $\ge 2$ and arbitrary Dedekind rings of arithmetic type. Also, we prove bounded generation of $\mathrm{St}(\Phi,\mathbb F_{q}[t,\,t^{-1}])$ for all root systems $\Phi$, and bounded generation of $\mathrm{St}(\Phi,\mathbb F_{q}[t])$ for all root systems $\Phi\neq\mathsf A_1$. The proofs are based on a theorem on bounded elementary generation for the corresponding Chevalley groups, where we provide uniform bounds.