Constructing Directed Cayley Graphs of Small Diameter: A Potent Solovay-Kitaev Procedure

TL;DR

This paper develops explicit bounds and a fast construction algorithm for directed Cayley graphs of finite quotients of groups, inspired by the Solovay-Kitaev procedure, with applications to specific algebraic groups.

Contribution

It introduces a novel method for bounding diameters and constructing directed Cayley graphs using a procedure analogous to Solovay-Kitaev, applicable to various groups.

Findings

Established diameter bounds for finite quotients of specific groups.

Provided a fast algorithm for constructing small-diameter directed Cayley graphs.

Applied the method to groups like SL_2 over formal power series and automorphism groups of rooted trees.

Abstract

Let $\Gamma$ be a group and $(\Gamma_n)_{n=1} ^{\infty}$ be a descending sequence of finite-index normal subgroups. We establish explicit upper bounds on the diameters of the directed Cayley graphs of the $\Gamma/\Gamma_n$ , under some natural hypotheses on the behaviour of power and commutator words in $\Gamma$. The bounds we obtain do not depend on a choice of generating set. Moreover under reasonable conditions our method provides a fast algorithm for constructing directed Cayley graphs of diameter satisfying our bounds. The proof is closely analogous to the the Solovay-Kitaev procedure, which only uses commutator words, but also only constructs small-diameter undirected Cayley graphs. As an application we give directed diameter bounds on finite quotients of two very different groups: $SL_2 (\mathbb{F}_q [[t]])$ (for $q$ even) and a group of automorphisms of the ternary rooted tree introduced by Fabrykowski and Gupta.