140 problems

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…

Let $d,k,n\in\mathbb{N}$, let $T_1,\dots,T_d:X\to X$ be invertible measure-preserving transformations on a probability space $(X,\mathcal{B}(X),\mu)$ that generate a nilpotent grou…

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

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…

Lipsman's conjecture. The CI condition is equivalent to properness of the pair $(L,H)$.

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 $G$ be a nilpotent group with complex Mal'cev completion, and let $\operatorname{RF}_G$ denote its residual finiteness growth. Mal'cev-completion invariance conjecture. The fun…

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

Let $G$ be a finite nilpotent group, let $w$ be a word in $k$ variables, and let $N_w(1)$ denote the number of tuples $(g_1,\ldots,g_k)\in G^{(k)}$ such that $w(g_1,\ldots,g_k)=1$.…

Bouc's conjecture. $\beta(G)$ is nilpotent if and only if $G$ is nilpotent.

Let $\Gamma$ be a finitely generated group. An injective endomorphism $\varphi\colon \Gamma\to\Gamma$ is strongly scale-invariant if $\varphi(\Gamma)$ has finite index in $\Gamma$…

Nilpotence conjecture. The root groups of every proper Moufang set are nilpotent.

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…

Burden formula.

Let $G$ and $H$ be simply connected nilpotent Lie groups. Two groups are quasiisometric when their underlying metric spaces are quasi-isometric. Quasiisometric rigidity conjecture.…

Degree conjecture. For every relevant $n$,

Parshin's conjecture. An irreducible representation of $G$ is monomial if and only if it has finite weight.

Let $\Gamma$ be a discrete group, and let $\rho_1$ and $\rho_2$ be left-invariant word metrics on $\Gamma$. Assume that … as $\gamma\to\infty$. Burago–Margulis bounded-distance con…

Let $R={\mathbb Z}[t]$ or $R={\mathbb Z}[t,t^{-1}]$, and let $K(R)$ be the kernel of the evaluation map $SL_n(R)\to SL_n({\mathbb Z})$. Pseudo-nilpotence conjecture. The group…

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

Let $G$ be a finite insoluble group satisfying the condition $(\mathcal{N},21)$, meaning that $G$ has 21 elements such that every pair generates a non-nilpotent subgroup. A finite…

Let $E_n$ be the variety of groups satisfying the identity $e_n(x,y)\equiv 1$, and let $F^n_k$ be the free group with $k$ free generators in this variety. Non-nilpotence conjecture…

Let $\Gamma$ be a finitely generated nilpotent group. Its automorphism tower is the sequence … where $\operatorname{Aut}^0 \Gamma=\Gamma$ and…

Nilpotency conjecture.

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