Experiments with the Markoff Surface

Abstract

We confirm, for the primes up to 3000, the conjecture of Bourgain-Gamburd-Sarnak and Baragar on strong approximation for the Markoff surface $x^2+y^2+z^2=3xyz$ 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 $x^2+y^2+z^2-3xyz=k$. In the degenerate case of the Cayley cubic, we give a complete description of the orbits.

Keywords:

• graphs and groups
• expander graphs
• strong approximation
• cubic surfaces
• Kesten-McKay law

Acknowledgments

We thank Peter Sarnak for his advice, encouragement, and support over the course of our work. We thank Pedro Henrique Pontes for showing us Lemma 2.2, which provides a simpler way to count solutions than our original proof using Gauss sums. We thank ReMatch, a summer research program at Princeton University, for being a supportive and stimulating research community. We thank the anonymous referee for reading the article and offering helpful suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Lee was supported by the Bershadsky Family Summer Research Scholars Fund through ReMatch and the Office of Undergraduate Research at Princeton University. de Courcy-Ireland was supported by a PGS D grant from the Natural Sciences and Engineering Research Council of Canada [PGSD2-471570-2015].