6 problems

Let $K=\mathbb{F}_p(t)$), let $T$ be the complete infinite rooted binary tree, and let $f_t=x^2+t\in K[x]$. Write $G_{f_t}(0)$ for the arboreal Galois group associated with the orb…

Let $T$ be the complete infinite rooted binary tree, and let $G\leq\operatorname{Aut}(T)$ be level-transitive if it acts transitively on every level. Define its Hausdorff dimension…

Let $R$ be a commutative pro-$p$ ring and let $G\le \mathrm{GL}_n(R)$ be a linear group over $R$. The group $W_p$ is the group of $p$-adic automorphisms, namely a Sylow pro-$p$ sub…

Let $f(x)\in\mathbb{Q}[x]$ be a quadratic polynomial, let $\alpha\in\mathbb{Q}$, let $T$ be the binary tree, and let … be the associated arboreal representation. An element of its…

Let $\phi_c(x)=x^2+c$ with $c\in\mathbb{Z}$. Write $G_n(\phi_c)$ for the Galois group at the $n$th stage of the arboreal representation and $G_\infty(\phi_c)$ for its full image, v…

For $a\in\mathbb{Z}$, let $\phi_a(x)=x^2+a$, let $T_n$ denote the rooted binary tree of height $n$, let $T_\infty$ denote the infinite rooted binary tree, and let $G_n(\phi_a)$ and…