96 problems

Atiyah's conjecture. The rank gradient $\beta_0^G(M)$ is an integer. This is a pro-$p$ analogue of the Atiyah conjecture concerning integrality of $L^2$-Betti numbers. The supplied…

Efrat's arithmetical Elementary Type Conjecture. There is such a decomposition where, for each $1\leq i\leq n$, one of the following holds: $G_i\cong\mathbb{Z}_p$; $G_i$ is the dec…

Let $p$ be an odd prime, let $K$ be a number field, and let $S$ be a finite set of tame primes of $K$. For a set $T$ of primes disjoint from $S$, let $G_{K,S}^T$ denote the Galois…

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…

Let $R$ be a pro-$p$ ring, and let $G$ be a pro-$p$ group that is linear over $R$. Linearity conjecture. The group $G$ cannot contain a non-abelian free pro-$p$ subgroup. This exte…

Shalev's conjecture. If $G$ satisfies some probabilistic identity $w$, then $G$ satisfies some identity $\tilde{w}$.

Fix an odd prime number $p$ and let $K$ be an imaginary quadratic field. Let $G_K$ be the Galois group of the maximal unramified pro-$p$ extension of $K$, called the $p$-tower grou…

Ozaki's non-freeness conjecture. The group $\mathcal{G}(k^\mathrm{c}_\infty)$ is not a non-abelian free pro-$p$-group.

Let $p$ be a prime and let a uniform pro-$p$-group mean a uniform pro-$p$ group. A group $G$ satisfies the unique product property (UPP) if, for every pair of nonempty finite subse…

Kernel Unipotent Conjecture. The Zassenhaus filtration of $G_K$ is given by the kernels of unipotent representations.

Let $\Gamma$ be a torsion-free abstract group, and let $\operatorname{rk}_{\Gamma}$ denote its associated rank function. The Atiyah conjecture. The value of…

Let $p$ be a prime, let $\mathbb{K}$ be a field containing a root of unity of order $p$, and, if $p=2$, suppose that $\mathbb{K}$ also contains $\sqrt{-1}$. Let $G$ denote the rele…

Let $p$ be a prime. A pro-$p$ group is of elementary type if it belongs to the class generated from $\mathbb{Z}_p$ and Demushkin groups by the constructions specified in the source…

Let $\mathcal{G}=(G,\theta)$ be a 1-smooth pro-$p$ pair, where $G$ is a pro-$p$ group and $\theta\colon G\to\mathrm{GL}_1(\mathbb{Z}_p)$ is a representation. The group $G$ is weakl…

Let $P$ be a pro-$p$ group, and write … Let $\varphi^2:H_{\operatorname{cont}}^{2}(P)\to H_{\operatorname{disc}}^{2}(P)$ be the comparison map. Prasad's conjecture. For every $p$-a…

Let $K=\mathbb{Q}_q$ be unramified over $\mathbb{Q}_p$, let $\overline{k}$ be an algebraic closure of its residue field, and let $P=\operatorname{Gal}(K^s/K^\mathrm{tr})$ be the wi…

Let $p>2$ be a prime and let $F$ be a field containing a primitive $p$-th root of unity. Write $F(p)$ for the maximal $p$-extension of $F$, let $G=\operatorname{Gal}(F(p)/F)$, and…

Let $R$ be a commutative pro-$p$ ring, let $G\subseteq GL_n(R)$ be a linear group over $R$, and let $\Gamma(p)$ be the profinite automorphism group of the rooted $p$-ary tree. The…

Let $\Gamma(p)$ be the profinite automorphism group of the rooted $p$-ary tree, and let positive-dimensional mean having positive Hausdorff dimension as a closed subgroup of…

A just infinite pro-$p$ group is a pro-$p$ group in which every proper quotient is finite, and a branch group is a just infinite pro-$p$ group of branch type. A nonabelian free pro…

Let $G$ be a just-infinite pro-$p$-group of finite width, meaning a just-infinite pro-$p$-group whose finite-index subgroup growth layers have uniformly bounded width. Zel'manov's…

Open Golod–Shafarevich subgroup conjecture. The group $\operatorname{Gal}(L/K)$ contains an open Golod–Shafarevich pro-$p$ subgroup.

Boston's virtually Golod–Shafarevich conjecture. The group $\operatorname{Gal}(L/K)$ contains an open pro-$p$ subgroup $G$ such that $d(G)$ and $r(G)$ are finite and

Reduced Boston conjecture. 1. If $G$ is a uniform $p$-adic analytic pro-$p$ group of simple type in characteristic zero and is hereditarily just-infinite, then there is no pro-$p$…

Demuškin maximal pro-$p$ Galois group conjecture. If $G_F(p)$ is a pro-$p$ Demuškin group of rank at least $3$, then $F$ is arithmetically Demuškin.