Strong boundedness of simply connected split Chevalley groups defined over rings

Abstract

This paper is concerned with the diameter of certain word norms on S-arithmetic split Chevalley groups. Such groups are well known to be boundedly generated by root elements. We prove 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. This property, called strong boundedness, was introduced by Kedra, Libmann and Martin and proven for ${\rm SL}_n(R)$, assuming R is a principal ideal domain and $n\geq 3$. We also provide examples of normal generating sets for S-arithmetic split Chevalley groups proving our bounds are sharp in an appropriate sense and give a complete account of obstructions to the existence of small normally generating sets of ${\rm Sp}_4(R)$ and $G_2(R)$. For instance, we prove that ${\rm Sp}_4(\mathbb{Z}[\frac{1+\sqrt{-7}}{2}])$ cannot be generated by a single conjugacy class.