5 problems

Markov group containment conjecture. For every $n\geq 1$, the associated Markov group is contained in the corresponding Galois group:

Let $k$ be a number field and let $f(x)\in k(x)$ be a quadratic rational function. The associated arboreal representation has image in the automorphism group of a regular rooted bi…

Local-to-global conjugacy conjecture. If $H\leq\operatorname{Aut}(\mathcal{T}_n)$ is locally conjugated into $M$, then $H$ is globally conjugated into $M$.

Let $k$ be a number field, and let $\phi(z)\in k(z)$ be a rational function of degree $d\geq 2$, defined over $k$. Let $L=k(t)$ be the function field, and define the generic arbore…

Let $K$ be a number field, let $f(z)=z^2-1$, and let $G_{\infty}$ be the associated arboreal Galois group for $f$ rooted at $x_0\in K$. Let $M_{\infty}$ be the arithmetic basilica…