Bounded generation of Chevalley groups, and around

We state several results on bounded elementary generation and bounded commutator width for Chevalley groups over Dedekind rings of arithmetic type in positive characteristic. In particular, Chevalley groups of rank $\ge 2$ over polynomial rings ${\mathbb F}_q[t]$ and Chevalley groups of rank $\ge 1$ over Laurent polynomial ${\mathbb F}_q[t,t^{-1}]$ rings, where ${\mathbb F}_q$ is a finite field of $q$ elements, are boundedly elementarily generated. We sketch several proofs, and establish rather plausible explicit bounds, which are better than the known ones even in the number case. Using these bounds we can also produce sharp bounds of the commutator width of these groups. We also mention several applications (such as Kac–Moody groups and first order rigidity) and possible generalisations (verbal width, strong bounded generation, etc.)
The talk is based on a joint work with Boris Kunyavskii, Eugene Plotkin.