Bounded elementary generation of Chevalley groups

We discuss several results on bounded elementary generation and bounded commutator width for Chevalley groups over Dedekind rings of arithmetic type in positive characteristics. In particular, Chevalley groups of rank $\ge 2$ over polynomial rings $F_q[t]$ and Chevalley groups of rank $\ge 1$ over Laurent polynomial $F_q[t,t^{-1}]$ rings, where $F_q$ is a finite field of $q$ elements, are boundedly elementarily generated. We sketch several proofs, using reciprocity laws, symbols in algebraic K-theory, and surjective stability for K-functors. As a result, we establish rather plausible explicit bounds, that do not depend on q and 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 (Kac—Moody groups, first order rigidity, etc.) and possible generalisations. This is joint work with Boris Kunyavskii and Eugene Plotkin.