141 problems

Positselski's conjecture. Absolute Galois groups of fields containing a $p$-th root of unity satisfy the Koszul property at $p$.

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…

A group is product separable if, for every finite collection of finitely generated subgroups $H_1,\ldots,H_k$, the product $H_1\cdots H_k$ is closed in the profinite topology. Pin–…

Let $k$ be a number field, let $\tilde{\mathbf{G}}$ be the ambient algebraic group, and let $\Gamma\subseteq\tilde{\mathbf{G}}(k)$ be an arithmetic subgroup. Write $V_{\infty}(k)$…

Let $K$ be a field containing an algebraically closed field, let $G_K$ denote its absolute Galois group, and let $p$ be a prime. Write $[G_K,G_K]$ for the commutator subgroup of…

Mann's conjecture. If $G$ is PFG, then $P_G(s)$ is absolutely convergent on some complex half-plane of $\mathbb C$ and, for sufficiently large integers $t$, satisfies

Let $k$ be a global field, let $S$ be a finite set of places of $k$, and let $\mathfrak{G}$ be a simple simply connected algebraic group over $k$. The $S$-arithmetic group…

Serre's congruence subgroup property conjecture. Every higher rank group has the congruence subgroup property (CSP), meaning

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 $G$ be a pro-$p$ group, and let $\operatorname{hspec}^{\mathcal G}(G)$ denote its Hausdorff spectrum with respect to a filtration $\mathcal G$. A pro-$p$ group is $p$-adic anal…

Profinite volume conjecture. The profinite completion of the fundamental group of a finite-volume hyperbolic $3$-manifold determines its hyperbolic volume.

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…

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 $\Gamma$ be a finitely generated residually finite group satisfying a non-trivial group law. The rank gradient $RG(\Gamma)$ measures the asymptotic normalized growth of the min…

Let $G$ be a group, and let $\Sigma_d(G)$ be the graph whose vertices are the ordered generating $d$-tuples of $G$, with two vertices adjacent if and only if they differ in exactly…

Open-subgroup triviality conjecture. If

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

Let $n$ be a positive integer. Every finite simple group of order $n$ has a profinite presentation with $2$ generators and at most $2\lambda(n)$ relations, where, if…

Let $p$ be a prime and $r$ a positive integer. The graph $\mathcal{G}(p,r)$ has finitely many components, and, after removing finitely many sporadic groups, its components are fini…

Let $G_{1},G_{2},\ldots,G_{n}$ be finite groups, let $s=\max_{i}d(G_{i})$, and let … be their free product. Write $\widehat{\Gamma}$ for the profinite completion of $\Gamma$, and l…

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

Let $F_d$ be the free group on $d$ generators, and let $n$ be a positive integer. Mann–Pyber conjecture. The number of normal subgroups of index $n$ in $F_d$ is … for some constant…

Let $F$ be a field with $\lvert F^\times/F^{\times 2}\rvert\le 8$, and consider possible Galois pro-$2$ groups with three generators arising from absolute Galois groups. Elementary…

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…