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

Diameters of Chevalley groups over local rings

Let G be a Chevalley group scheme of rank l. We show that the following holds for some absolute constant d>0 and two functions p_0=p_0(l) and C=C(l,p). Let p>p_0 be a prime number and let G_n:=G(\Z/p^n\Z) be the family of finite groups for n>0. Then for any n>0 and any subset S which generates G_n we have diam(G_n,S)< C n^d, i.e., any element of G_n is a product of Cn^d elements from S\cup S^{-1}. In particular, for some C'=C'(l,p) and for any n>0 we have, diam(G_n,S)< C' log^d(|G_n|). 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, there exists an efficient algorithm to compute a short path between any two vertices in any Cayley graph of the groups G_n.Comment: 8 page

Groups and Geometries

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