21 problems

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…

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…

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

Limit-exponent conjecture. Prove that there exists a limit

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 $K$ be a finitely presented group with growth function $N_K(L)$, where $N_K(L)$ counts the elements in the radius-$L$ ball of a Cayley graph. Groups with growth intermediate be…

Let $G$ be a finitely generated virtually nilpotent group with symmetric generating set $S$, and let $S(g,n)$ denote the sphere of radius $n$ about $g$. Suppose that $G$ has polyno…

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

Let $G$ be a finitely generated group with a finite generating set $X$. Its degree of commutativity with respect to $X$ is … where $B_{G,X}(n)$ is the ball of radius $n$ in the Cay…

Let $BS(p,q)=\langle a,t\mid t a^p t^{-1}=a^q\rangle$, and consider the growth of its horocyclic subgroup $\langle a\rangle$ with respect to the induced metric from the standard ge…

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…

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…

Volume-error exponent conjecture. One should have

Let $G$ be a finitely generated group with a finite generating set $X$, and let $\gamma_G(n)$ denote its ordinary growth function. A function $f$ is exponential if $f\sim 2^n$ and…

Let $S$ be a generating set in the setting of the theorem above, whose conclusion gives a bound involving a constant $C_d$ and an implied multiplicative constant, with $d$ denoting…

For a finitely presented group $G$ with deficiency $d$, where deficiency means that $G$ has a presentation with $n$ generators and $n-d$ relators, let $omega(G)$ denote its uniform…