Baragar–Bourgain–Gamburd–Sarnak strong approximation conjecture for the Markoff equation

Let

$$\mathbb{X}: x^2+y^2+z^2-xyz=0$$

be the Markoff equation, and for a prime $p$ write

$$\mathbb{X}^*(p):=\mathbb{X}(\mathbb{F}_p)\setminus\{(0,0,0)\}.$$

The equation satisfies strong approximation at $p$ when the reduction map

$$\mathbb{X}(\mathbb{Z})\longrightarrow\mathbb{X}(\mathbb{F}_p)$$

is surjective.

Baragar–Bourgain–Gamburd–Sarnak conjecture. The Markoff equation $\mathbb{X}$ satisfies strong approximation at every prime $p$. Equivalently, $\operatorname{Out}(\Pi)$ acts transitively on $\mathbb{X}^*(p)$ for every prime $p$.

Strong approximation is equivalent here to the connectedness of the corresponding moduli fiber and describes the abundance of integral points on the Markoff surface. The conjecture is attributed to Baragar and was reaffirmed by Bourgain, Gamburd, and Sarnak; the paper records effective bounds for the possible exceptional primes.