Problem of the Week
Hilton Memorial Lecture
This talk will be on grouph [sic] theory.
Given a finite group G, I examine various interesting graphs with vertex set Sn(G) of generating n-tuples of G, most importantly the (extended) Product Replacement Graphs and the Andrew-Curtis Graphs. The Product Replacement Graphs are of great interest as suitable random walks on these graphs produce excellent ways of generating random elements of the group G. These graphs admit interesting actions of both Aut(Fn) (Fn is the free group) and Aut(G) motivated by combinatorial and computational group theory. In particular I will discuss connectivity issues related to these graphs in some generality before focusing on what is already known for solvable groups and certain families of simple groups.