Bounded generation of Steinberg groups over Dedekind rings of arithmetic type

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.