Strong boundedness of split Chevalley groups (Q2696631)

A group \(G\) is called boundedly generated by a set \(S \subseteq G\), if there is a natural number \(N\) such that \(G=(S^{-1}S)^{N}\). If \(g \in G\) the word norm \(||g||_{S}\) is the smallest number of conjugates of elements of \(S^{-1} \cup S\) needed to write \(g\). This paper is concerned with the diameter of certain word norms on \(S\)-arithmetic split Chevalley groups, which are known to be boundedly generated by root elements. The author proves that word metrics given by conjugacy classes on \(S\)-arithmetic split Chevalley groups have an upper bound only depending on the number of conjugacy classes. He also provides examples of normal generating sets for \(S\)-arithmetic split Chevalley groups proving that its bounds are sharp in an appropriate sense and gives a complete account of the existence of small normally generating sets of \(Sp_{4}(R)\) and \(G_{2}(R)\) assuming \(R\) is a principal ideal domain.