14 problems

Develin–Hartke conjecture. There exists an outbreak on $\mathbb{Z}^d$ which cannot be contained by deploying $f(n)$ firefighters at time $n$.

Commuting infinite subgroups conjecture. The group $G$ contains two infinite subgroups $H_1$ and $H_2$ which commute with each other:

Finitely presented intermediate-growth conjecture. There are no finitely presented groups of intermediate growth.

Positive-power lower-bound conjecture. Then

Free-subgroup conjecture. The group $G^2_4$ has a free subgroup of rank $2$.

Let $G$ be a $\det$-finite group, meaning that the relevant Fuglede–Kadison determinants are finite, and let $\Phi\colon G\to G$ be an automorphism. Assume that $G$ has subexponent…

Let $G$ be a finitely generated group of super-polynomial growth, and let ${}^{e}\operatorname{pdim}(G)$ denote its exponential percolation dimension. Positive-dimension conjecture…

Let $G$ be a finitely generated group, and let ${}^{e}\operatorname{pdim}(G)$ denote its exponential percolation dimension. Gap conjecture. There is a $\beta>0$ such that … for eve…

Let $G$ be a finitely generated group of exponential word growth, and let $S$ be a finite generating set of $G$. Antolín–Martino–Ventura conjecture. Then … irrespective of the choi…

Conjugacy-ratio conjecture. The group $G$ has positive conjugacy ratio if and only if $G$ is virtually abelian:

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 $G$ be a finitely generated group, and let $X$ be a finite generating set. Define the degree of commutativity with respect to $X$ by … Here $\mathbb{B}_X(n)$ is the ball of rad…

Given $d\in\mathbb{N}$, let $S$ be a finite subset of $\operatorname{GL}_d(\mathbb{C})$ and define its exponential growth rate by … where $S^n$ is the set of products of $n$ elemen…

Let $G$ be a simply connected nilpotent Lie group, let $\Gamma$ be a discrete co-compact subgroup, and let $S$ be a finite symmetric generating set containing the identity. Let…