Diameters of Chevalley groups over local rings

Let G be a Chevalley group scheme of rank l. Let ${G_n := G(\mathbb{Z} / p^{n} \mathbb{Z})}$ be the family of finite groups for ${n \in \mathbb{N}}$ and some fixed prime number p >?p ₀. We prove a uniform poly-logarithmic diameter bound of the Cayley graphs of G _n with respect to arbitrary sets of generators. In other words, for any subset S which generates G _n , any element of G _n is a product of C n ^d elements from ${S \cup S^{-1}}$ . Our proof is elementary and effective, in the sense that the constant d and the functions p ₀(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 G _n .