We 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.
Athanasios Nenes, Paraskevi Georgakaki
Paolo De Los Rios, Pierre Goloubinoff, Satyam Tiwari, Mathieu Rebeaud, Bruno Claude Daniel Fauvet, Adélaïde Alice Mohr
Andreas Mortensen, David Hernandez Escobar, Léa Deillon, Alejandra Inés Slagter, Eva Luisa Vogt, Jonathan Aristya Setyadji