66 problems

Central-quotient eigengap conjecture. Either $k_{>1}=|G|-1$, or

Let $p$ be a prime, and suppose that $G=P\times A$ is a finite group, where $P\in\operatorname{Syl}_p(G)$ is non-abelian and $A$ is an abelian $p'$-group. Let $\Gamma_K$ denote the…

Let $G$ be a finite non-abelian nilpotent group, and let $H$ be a group. For a finite non-abelian group $K$, let $\Gamma_K$ be the graph whose vertices are the non-central elements…

Scaling conjecture. There is a map $\alpha:S^k\to G$ such that

Nilpotency conjecture.

Let $X$ be a normal complex algebraic variety. Suppose that $\pi_1(X)$ is virtually nilpotent. Campana's conjecture. Then $\pi_1(X)$ is virtually at most $2$-step nilpotent. This c…

Bergelson–Leibman conjecture. The limit of these averages exists in $L^2$-norm and almost everywhere. The conjecture concerns pointwise and norm convergence of polynomial multiple…

Guo–Tointon conjecture. $G$ has the u.c. property if and only if it is virtually nilpotent.

2-step nilpotency conjecture. The group $\pi_1(X)$ is virtually nilpotent of class at most $2$.

Furstenberg–Bergelson–Leibman conjecture. These averages converge for $\mu$-almost every $x\in X$ as $\min\\{N_1,\ldots,N_k\\}\to\infty$. When $N_1=\cdots=N_k=N$, write the average…

The LEF nilpotent-product conjecture. For each $k\geq 1$, the $k$-nilpotent product of a finite family of LEF groups is LEF.

Let $G$ be a torsion-free nilpotent group, and let $\delta$ be the constant in the previously established upper bound for the residual finiteness growth $\operatorname{RF}_G$. Exac…

Let $M$ be a compact Riemann manifold with negative curvature. Let $\Gamma$ be a finitely generated torsion-free nilpotent group, let $\Phi:\pi_1(M)\to\Gamma$ be surjective, and le…

Let $\varphi_t$ be a smooth, transitive and weakly mixing Anosov flow on a compact manifold $N$. Let $\Gamma$ be a finitely generated torsion-free nilpotent group, let…

Vdovin's conjecture. Then $H \cap H^x = 1$ for some $x \in G$.

Let $G$ be a finite $p$-group. Denote by $\gamma_i(G)$ the $i$th term of the lower central series of $G$, by $Cl(G)$ the generalized class parameter used in the paper, and by…

Let $G$ and $H$ be simply connected nilpotent Lie groups. Losert–Malcev conjecture. The groups $G$ and $H$ are quasi-isometric if and only if they are isomorphic. This would mean t…

Let $\mathrm{S}_n$ be the symmetric group on $n$ points. A random subgroup of $\mathrm{S}_n$ means a subgroup chosen uniformly from the finite set of all subgroups of…

Wilson's conjecture. Every locally nilpotent omega-categorical $p$-group is nilpotent.

Let $G$ be a finitely generated residually finite group. For $d e 0$, write uniformly $(1/d)$-almost flat quotients and uniformly $(1/d)$-almost flat coset spaces for the correspon…

For $k\geq 2$, let $\sum_{<k}$ be the collection of words in $*_1,\ldots,*_d$ in which each letter occurs at most $k-1$ times. For elements $a,x_1,\ldots,x_d$ of an $s$-step nilpot…

Free two-step nilpotent Busemann-point conjecture. The set of Busemann points of $N_{2,r}$ with respect to the standard generating set is countable.

Let $G$ be a finitely generated group. A group is strongly scale-invariant if it admits an injective homomorphism $f:G\to G$ such that $f(G)$ is a proper finite-index subgroup of…

Let $G$ be a group and let $x\in G$. Suppose that the nilpotentizer $nil_G(x)$ has order $8$ and generates a maximal subgroup of $G$. Solvability conjecture. Then $G$ is solvable.…

Let $K$ and $L$ be simply connected nilpotent Lie groups with one-dimensional centres, of nilpotency classes $k$ and $\ell$, respectively, and let their central product be … Here…