Diameters of Chevalley groups over local rings

Abstract

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 .