Connectivity of the product replacement algorithm graph of PSL(2, q)

Abstract

The product replacement algorithm is a practical algorithm to construct random elements of a finite group G . It can be described as a random walk on a graph Г k ( G ) whose vertices are the generating k -tuples of G (for a fixed k ). We show that if G = PSL(2, q ) or PGL(2, q ), where q is a prime power, then Г k ( G ) is connected for any k 4. This generalizes former results obtained by Gilman and Evans.