12 problems

Bourgain–Gamburd–Sarnak connectivity conjecture. The graph ${\mathcal{G}(p)}$ is connected for every prime $p>3$. This is the strong-approximation conjecture for the Markoff equati…

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

Let $p$ be a prime, and let $\mathcal{G}_p$ be the Markoff graph over $\mathbb{F}_p$: its vertices are the nonzero triples in $\mathbb{F}_p^3$ satisfying … and two vertices are joi…

Let $\mathcal M$ be the set of positive integer solutions to … and let $\mathcal M_p$ be the set of non-zero solutions modulo a prime $p$. The functional graph is generated by the…

Let $\mathbf{M}$ be the modular torus, the hyperbolic once-punctured torus described in the paper. A pair of simple closed geodesics is a pair of simple geodesics on $\mathbf{M}$ w…

Consider positive integer solutions to the Markoff cubic … For two solutions ordered as $x\geq y\geq z$ and $x\geq y'\geq z'$, Markoff's conjecture. One has $y=y'$ and $z=z'$. This…

Let $a,b,c$ be positive integers satisfying the Markoff equation … Order each triple so that $a\leq b\leq c$. Frobenius's Markoff conjecture. The set of such triples is uniquely de…

A Markoff triple is a positive integer solution $(a,b,c)$ of the Markoff equation … Order its entries so that $a\leq b\leq c$. Frobenius unicity conjecture. The Markoff triple…

Density conjecture. Through numerical experimentation, the limit $\delta$ exists and is approximately $40\%$. The preceding theorem gives the upper bound $\delta<0.5$, while the as…

Let $p$ be a prime and consider the Markoff equation … A solution modulo $p$ is a triple $(X,Y,Z)\in\mathbb{F}_p^3$ satisfying this equation over $\mathbb{F}_p$; it lifts to an int…

Let $\mathcal{M}$ be the set of Markoff triples over $\mathbb{Z}$, let $\mathcal{M}_p$ be the set of their reductions modulo a prime $p$, and let $\mathcal{G}^\times_p$ be the Mark…

Let $G_M$ be the graph whose vertices are the positive Markoff numbers, with edges arising from the Markoff transformations between coordinates of solutions to $x^2+y^2+z^2=3xyz$.…