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Markov numbers and trees of traces

The starting point of our talk is the intimate connection between Markov numbers, that is, the integer solutions of the equation x^2 + y^2 + z^2 = 3xyz, and the free group F_2 on two generators. In fact there is a binary tree of traces associated to any representation of of F_2 into SL(2,C), of which Markov numbers are a particular example. Brian Bowditch found a way to use this tree to give a purely combinatorial proof of McShane's identity, one of the building blocks of Mirzakhani's work on simple curves and moduli spaces. Extensions of his ideas lend themselves to computer explorations which have produced some interesting new results but also leave a number of unanswered questions.