10 problems

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…

Let $P$ be a postcritically-finite non-renormalizable quadratic polynomial whose Julia set $\mathcal{J}_P$ is a dendrite. Intermediate-growth conjecture. The iterated monodromy gro…

Let $\ThePolynomial=\TheComplex^2+\TheConstTerm$ be a quadratic polynomial, and let $\operatorname{IMG}(\ThePolynomial)$ denote its iterated monodromy group acting on the infinite…

Let $f\in \mathbb{C}[x]$ be a post-critically finite rational function. Write $\mathrm{IMG}(f)$ for its iterated monodromy group, and say that it has $(\mathrm{LR})$ when it has th…

Let $f$ be a dynamically exceptional polynomial, and suppose that $f$ is not conjugated to a Chebyshev polynomial. Write $\operatorname{IMG}(f)$ for its iterated monodromy group an…

Finite-type recurrence conjecture. The group $G(f)/k$ is a recurrent group of finite type generated by some pattern group $\mathcal{P}$ of depth $d$, where

Let $f(x)=1/(x-1)^2$, and let $G_n^{\mathrm{geom}}(f)$ denote its geometric iterated monodromy group at level $n$. Conjecture 7.15. For the relevant levels $n$, one has … This conj…

Let $f: \mathbb{C}^n \to \mathbb{C}^n$ be post-critically finite. An affine Weyl iterated-monodromy characterization conjecture asserts that if the iterated monodromy group of $f$…

Let $H_q$ be the Thue–Morse group, realized as the iterated monodromy group of the degree-$q$ branched covering described using the post-critical set…

Let $\Gamma$ be the iterated monodromy group of any postcritically finite quadratic morphism … For each $w\in\Gamma$, let $\Phi_w$ denote its orbit length generating function. Rati…