Mathematical Conversations

Let G be a group, and let (g,h) be a pair in G x G. Consider the group of symmetries of G x G generated by the "moves" sending (g,h) to (g,gh), (g,g^{-1}h), (g,hg), (g,hg^{-1}), (gh,h),...etc. An old question from the 50's, motivated by the study of group presentations, asks: What are the orbits of this action? In algebraic geometry, such questions are related to the problem of determining the connected components of Hurwitz spaces. I will explain this connection, some recent progress, and an application to a conjecture of Bourgain, Gamburd, and Sarnak on the Diophantine properties of the Markoff equation x^2 + y^2 + z^2 - xyz = 0.