The study of Markoff triples, solutions to $ x^{2} + y^{2} + z^{2} = 3xyz $, spans over many fields. In this talk, we discuss arithmetic of Markoff triples by considering Markoff mod-p graphs. We will delve into what is known about their connectivity, going into a recent result which is joint work with Eddy, Litman, Martin, and Tripeny.
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