de Courcy-Ireland, MatthewLee, Seungjae2020-03-032020-03-032020-03-03202210.1080/10586458.2019.1702123https://infoscience.epfl.ch/handle/20.500.14299/166690WOS:000506504200001We confirm, for the primes up to 3000, the conjecture of Bourgain-Gamburd-Sarnak and Baragar on strong approximation for the Markoff surface modulo primes. For primes congruent to 3 modulo 4, we find data suggesting that some natural graphs constructed from this equation are asymptotically Ramanujan. For primes congruent to 1 modulo 4, the data suggest a weaker spectral gap. In both cases, there is close agreement with the Kesten-McKay law for the density of states for random 3-regular graphs. We also study the connectedness of other level sets . In the degenerate case of the Cayley cubic, we give a complete description of the orbits.Mathematicsgraphs and groupsexpander graphsstrong approximationcubic surfaceskesten-mckay lawprimetext::journal::journal article::research article