Joint IAS/PU Groups and Dynamics Seminar

The generalized Markoff equation gives rise to a dynamical system via the Markoff group acting on the solution set. This talk considers the resulting dynamics over finite fields, which has strong connections with combinatorial group theory and number theory. Specifically, we focus on the orbit structure of the Markoff group action, which McCullough and Wanderley conjecture to be trivial outside of a small number of exceptional orbits. Cases of this conjecture have been verified in recent work of Chen and of Bourgain–Gamburd–Sarnak. We will present new techniques that establish the McCullough–Wanderley conjecture in additional settings.