Diameters of Chevalley groups over local rings

TL;DR

This paper establishes explicit bounds on the diameters of Chevalley groups over local rings, demonstrating polynomial growth and providing an efficient algorithm for shortest path computation in their Cayley graphs.

Contribution

It provides explicit, effective bounds on diameters of Chevalley groups over local rings and introduces an efficient algorithm for shortest path computation.

Findings

Diameter bounds grow polynomially with the group size.

Explicit constants and functions for diameter bounds are derived.

An efficient algorithm for shortest path computation in Cayley graphs is presented.

Abstract

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.