21 problems

Let the graphs arising from the other surfaces in the paper's family be the analogous graphs obtained from the corresponding surface actions. Bourgain–Gamburd–Sarnak's expander con…

Baragar's transitivity conjecture. Just as $\Gamma$ acts transitively on $\mathcal{M}(\mathbb{Z}_{>0})$, it should act transitively on $\mathcal{M}^*(\mathbb{F}_p)$. This would imp…

Let $p$ be a prime, and let … be a Markoff surface over $\mathbb{F}_p$. Call a level $k$ generic when it contains no orbits associated with $A_4$, $S_4$, or $A_5$. Let…

Let $p$ be a prime and let $\mathbf{M}_k$ be the Markoff surface … Let $\mathcal{E}_k$ denote the union of the exceptional orbits in $\mathbf{M}_k(\mathbb{F}_p)$, and let…

Minimality conjecture. Outside the $p^{-1}$-neighborhoods of these periodic points, minimality holds over the $p$-adic integer points; equivalently, the relevant automorphism dynam…

For each prime $p$, let $\mathcal{G}_p$ be the corresponding Markoff modulo-$p$ graph. Super Strong Approximation. The collection of Markoff modulo-$p$ graphs $(\mathcal{G}_p)_p$ f…

Let $(\mathcal{G}_p)_p$ denote the family of Markoff modulo-$p$ graphs associated with the action of the Vieta group on the nonzero points of the Markoff surface over…

Let $\mathcal{X}$ be the affine surface in $\mathbb{A}^3$ defined by … and let $\Gamma$ be the group of affine integral morphisms generated by coordinate permutations and the Vieta…

Markoff graph expander conjecture. The graphs $\overline{\mathcal M}(\mathbb F_q)$ form a family of expander graphs. This conjecture underlies the use of Markoff graphs in cryptogr…

Baragar's conjecture. This reduction map is surjective for every prime $q$. Chen proved this for all sufficiently large primes, while the conjecture for all primes remains an inter…

For each integer $m$, let $\mathcal{U}_m$ denote the corresponding integral Markoff surface, and let $\mathbb{A}_{\mathbb{Z}}$ denote the adelic space used to formulate the integra…

For an integer $m$, let $U_m$ be the affine Markoff surface … For $m=0$, let $U_0^*(\mathbb{Z}/p\mathbb{Z})=U_0(\mathbb{Z}/p\mathbb{Z})\setminus\{(0,0,0)\}$. Let $\Gamma$ be the gr…

Let $a,k\in\mathbb Z$, and let … be the Markoff surface over $\mathbb Z$, with $\mathcal G_{\mathcal M}$ the group generated by the three involutions, double sign changes, and coor…

Let $\mathcal M_{3,0}$ be the affine Markoff surface … and let $\mathcal G_{\mathcal M}$ be the group of automorphisms generated by the three involutions, double sign changes, and…

For a prime $p$, let $M(\mathbb{F}_p)$ be the set of nonzero triples $(x,y,z)\in\mathbb{F}_p^3$ satisfying … and let $\mathfrak{M}_p$ be the graph on this vertex set with edges joi…

Let $\mathbb{X}$ be the affine surface defined by … For a prime $p$, let $X^{*}(p)$ be the set of nonzero solutions in $\mathbb{Z}/p\mathbb{Z}$, and partition it into blocks obtain…

Let $\mathbb{X}$ be the affine surface defined by … Let $\Gamma$ be the group generated by coordinate permutations and the Vieta involutions preserving this equation. For a prime…

Let $g\in\operatorname{PSL}_{2}(\mathbb{Z})$ be hyperbolic. Let $L(g;p)$ denote the longest orbit of $g$ on $\mathbb{Y}_{-2}(\mathbb{F}_{p})$, where…

Let $H(p)$ be the permutation group induced by the action of $\operatorname{Out}(\mathbf{F}_{2})$ on $\mathbb{Y}_{-2}(\mathbb{F}_{p})$, and let … Here…

Strong Approximation Conjecture. For any prime $p$, $\mathbb X(\mathbb Z/p\mathbb Z)$ consists of exactly two $\Gamma$-orbits, namely

Let … and let $\Gamma$ be the group of affine morphisms generated by coordinate permutations and the Vieta involutions preserving $X$. For a prime $p$, write…