Daniel Martin of Clemson University presents a research seminar talk, part of the joint IAS/Princeton University Groups and Dynamics Seminar, on the dynamics of the Markoff group acting on solutions to the generalized Markoff equation over finite fields. Martin lays out the connection between this dynamical system and combinatorial group theory and number theory, focusing on the orbit structure of the group action. He frames the talk around the McCullough-Wanderley conjecture, which predicts that this orbit structure is trivial except for a small number of exceptional cases, and surveys prior progress on the conjecture from Chen and from Bourgain, Gamburd, and Sarnak. The core of the talk is new techniques Martin and collaborators have developed to establish the conjecture in additional settings. The audience is specialists in number theory and group theory, and the talk assumes familiarity with algebraic varieties and finite field arithmetic rather than offering a general introduction.