74 problems

Let $w(x_1,\dots,x_k)$ be a word in $k$ variables, and let $G$ be a profinite group. Write $w\{G\}$ for the set of word-values of $w$ in $G$, and $w(G)=\langle w\{G\}\rangle$ for t…

Let $p$ be a prime, let $T_p$ be the rooted $p$-ary tree, and let $W_p$ be its full automorphism group. A pro-$p$ group is just-infinite if all its non-trivial closed normal subgro…

Open-subgroup triviality conjecture. If

Blaubeuren's conjecture. A profinite group cannot have an infinite finitely generated image.

Kionke's conjecture. The shifted partial Riemann zeta function $\zeta_S(s-1)$ has the same abscissa of convergence as the Weil representation zeta function $\zeta^W_{H_S}(s)$. This…

Mann's conjecture. The quantity $|\mu(H,G)|$ is bounded by a polynomial function of $|G:H|$, and the number of subgroups $H$ of $G$ of index $m$ satisfying $\mu(H,G)\neq 0$ grows a…

Hilbertian-projective implies pro-free conjecture. If $K$ is Hilbertian and projective, then $G_K$ is profree.

Let $G$ be an unbounded quasi-Engel group, meaning that for every $x,y\in G$ there is an integer $n=n(x,y)$ such that ${}^w\!u_n(x,y)=1$, for the fixed initial word $w$. A group is…

Let $\Gamma(p)$ be the profinite automorphism group of the rooted $p$-ary tree, let $G$ be a closed subgroup, and let Hausdorff dimension be computed in $\Gamma(p)$. A subgroup is…

Let $\Gamma(p)$ be the profinite automorphism group of the rooted $p$-ary tree, and let the closure of a subgroup generated by random elements be considered in the profinite topolo…

Projectivity conjecture. $H$ is projective.

Profinite Bateman–Horn conjecture. The following limits exist and satisfy

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…

A limit group is a finitely generated subgroup of a group obtained from a free group by finitely many extensions of centralizers, and it is one-ended when it has one end. For a gro…

Let $G$ be the group and $a$, $d$, and $P$ the element, rank, and set of primes occurring in conditions (a), (b), and (c) of the surrounding discussion. Baby Remeslennikov Conjectu…

Let $(t_n)_{n\geq 0}$ be a rapidly growing sequence of natural numbers, let $F_d$ be the free group, and construct a nested sequence of finite-index subgroups $(\Lambda_n)$ by taki…

Torsion conjecture. The quotient

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…

Profinite rigidity conjecture for word automorphic orbits. If there exists $\widehat{f}\in\operatorname{Aut}(\widehat{F}_k)$ such that

Profinite rigidity conjecture. Every cusped hyperbolic $3$-manifold is profinitely rigid among compact, orientable $3$-manifolds.

The Weil abscissa conjecture. For every set $S$ of primes,

Let $\mathcal{M}$ and $\mathcal{N}$ be hyperbolic $3$-manifolds. Their profinite completions are the inverse limits of their finite quotient groups. The manifolds are commensurable…

Let $\Sigma$ be the infinite family of finite non-abelian simple groups … Suppose that a profinite group $G$ satisfies $P_2(G)>0$, where $P_2(G)$ denotes the proportion of 2-elemen…

Mel'nikov's conjecture. An infinite and residually finite Mel'nikov group is either a free group, a surface group or a solvable Baumslag--Solitar group.

Let $N$ be a finite non-abelian characteristically simple group, let $a,b\in\operatorname{Aut}(N)$, and identify $N$ with $\operatorname{Inn}(N)$. Define…