Markoff Surfaces and Superstrong Approximation

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The existence of irrational numbers has fascinated mankind since antiquity; their approximation by rational numbers is of great importance in both theory and applications. The Markoff Diophantine equation arose in Markoff's fundamental work (in 1879) characterizing those irrationals that are badly approximable. The equation arises in several fields of mathematics. Its integer solutions have a tree structure, and investigation of the arithmetic properties of these Markoff numbers (the first of which is associated with the golden ratio, in a sense the most badly approximable irrational number) leads to important questions in graph theory. This research project investigates the connectivity of graphs related to the Markoff tree. In addition to its importance for number theory, this investigation has deep connections and applications to, among other topics, the product replacement algorithm (the most prevalent but still poorly understood tool in computational group theory) and expander graphs in computer science.Investigation of the arithmetic properties of Markoff numbers leads to the question of whether the graphs obtained by the modular reduction of the Markoff tree are connected (strong approximation). Superstrong approximation is the assertion that these graphs are in fact highly connected, that is to say form a family of expanders. The past decade saw a remarkable explosion of activity in the area of superstrong approximation for thin groups (Zariski dense subgroups of infinite index). Much of the research was driven by development of the affine sieve. In the case of thin groups with Levi factor of its Zariski closure semisimple, the strong and superstrong approximation and their applications in affine sieve are by now well-understood. On the other hand, the tori pose particularly difficult problems, both in terms of sparsity of elements in an orbit and their Diophantine properties as well as in terms of strong approximation, which in this case amounts to Artin's primitive root conjecture. This research project will investigate strong and superstrong approximation in a setting that is intermediate in level of difficulty between that of tori and that of thin linear groups, namely, that of nonlinear actions on a surface defined by the Markoff equation as well as in the context of other surfaces of Markoff type. Superstrong approximation results for thin linear groups will play an important role in this investigation, as will techniques and methods related to progress on Lang's conjecture and to the classification of algebraic Painlevé VI equations.