Bounding Lifts of Markoff Triples mod p*

Abstract: In 2016, Bourgain, Gamburd, and Sarnak proved that Strong Approximation holds for the Markoff surface in most cases. That is, the modulo p solutions to the equation $x^2+y^2+z^2=3xyz$ are covered by the integer points for most primes p. In this talk, we will discuss how the algorithm given in the paper of Bourgain, Gamburd, and Sarnak can be used to obtain upper bounds on lifts of Markoff triples modulo p. We will also discuss ongoing work to improve these bounds on average by assuming the Markoff mod p graphs form an expander family. This is joint work with Elena Fuchs and Lynnelle Ye.

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