# Experiments with the Markoff surface

## TL;DR

This paper verifies a conjecture about the Markoff surface for primes up to 3000, explores spectral properties of related graphs, and analyzes the structure of certain level sets, providing new empirical evidence and detailed classifications.

## Contribution

It confirms the Bourgain-Gamburd-Sarnak conjecture for primes up to 3000 and investigates spectral and structural properties of graphs and level sets associated with the Markoff surface.

## Key findings

- Data supports strong approximation conjecture for primes up to 3000.
- Graphs for primes ≡ 3 mod 4 are asymptotically Ramanujan.
- Spectral gaps vary for primes ≡ 1 mod 4, aligning with the Kesten-McKay law.

## Abstract

We confirm, for the primes up to 3000, the conjecture of Bourgain, Gamburd, and Sarnak 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.