Diameters of Chevalley groups over local rings

Let G be a Chevalley group scheme of rank l. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G_n := G(\mathbb{Z} / p^{n} \mathbb{Z})}$$\end{document} be the family of finite groups for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n \in \mathbb{N}}$$\end{document} and some fixed prime number p > p0. We prove a uniform poly-logarithmic diameter bound of the Cayley graphs of Gn with respect to arbitrary sets of generators. In other words, for any subset S which generates Gn, any element of Gn is a product of Cnd elements from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S \cup S^{-1}}$$\end{document}. Our proof is elementary and effective, in the sense that the constant d and the functions p0(l) and C(l, p) are calculated explicitly. Moreover, we give an efficient algorithm for computing a short path between any two vertices in any Cayley graph of the groups Gn.