790 problems

Let $\mathcal{G}$ be a graph on $n$ vertices with non-abelian automorphism group $G=\operatorname{Aut}(\mathcal{G})$ of order $n$, and let $\mathbf{x}$ be a single observation of a…

Converse Property R graph conjecture. For any configuration ${\cal G}$ of stars ${\cal S}_j$ and isolated points as in the characterization theorem, there exists a group $G$ such t…

Let $M$ be a closed aspherical $n$-manifold. Singer's conjecture. The $ell^2$-Betti numbers satisfy … for all $p\neq n/2$. The Singer conjecture is a major unresolved problem conce…

Determine whether every subcomplex $Y$ of every aspherical $2$-dimensional CW complex $X$ is aspherical; that is, whether $\pi_n(Y)=0$ for every $n\ge 2$ whenever $\pi_n(X)=0$ for…

Let $\mathcal{C}$ denote the set of class transpositions. Is the set $\{\operatorname{ord}(\tau_1\tau_2):\tau_1,\tau_2\in\mathcal{C},\ \operatorname{ord}(\tau_1\tau_2)<\infty\}$ fi…

For each $n\geq 4$, let $T_n=\{\mathrm{id}\}\cup\{(i\ j):1\leq i<j\leq n\}\subseteq S_n$. Is it true that there is no subset $A\subseteq S_n$ such that the multiplication map…

Let $G$ be a simple Lie group of real rank $r\ge 2$, let $\Gamma<G$ be a lattice, and let $X$ be an acyclic simplicial complex with $\dim X<r$. Then every simplicial action of…

Let $A_\Gamma$ be an Artin group and let $W_\Gamma$ be its associated Coxeter group, obtained from $A_\Gamma$ by imposing $s_i^2=1$ for every standard generator $s_i$. For a group…

For every integer $n\ge 2$, the involution length of $\mathrm{PU}(n,1)$ is exactly $4$: every $g\in\mathrm{PU}(n,1)$ can be written as a product of at most four nontrivial involuti…

Let $P$ be a group, let $L(P)$ denote its free multiplicative Lie algebra, let $\Gamma_n(P)$ be the normal subgroup generated by Lie-bracket commutators of weight $n$, and let…

For every word $U\in F(x_0,x_1)$, let $G_U=\langle x_0,x_1\mid U(x_0,x_1)U(x_1,x_0)^{-1}\rangle$. If $G_U$ is solvable and does not contain a free subgroup of rank $2$, then $G_U$…

For every finite group $G$, every integer $n\ge 1$, every choice of subgroups $H_1,\ldots,H_n\le G$, and every $g_1,\ldots,g_n\in G$, if $U=\bigcup_{i=1}^n g_iH_i$ is a proper subs…

For every connected semisimple Lie group $G$ with finite centre and every degree $n\ge 0$, the comparison map from bounded continuous cohomology to continuous cohomology,…

For every torsion-free discrete group $G$, the first $^2$-Betti number satisfies $\beta^{(2)}_1(G)\leq \operatorname{nrk}(G)-1$, where…

For every finite field $k$ of characteristic $p>0$ and every integer $n\ge 3$, the tame polynomial automorphism group $\operatorname{TA}_n(k)$ is not finitely generated; equivalent…

For every fixed integer $n\ge 2$, the family of undirected generalized pancake graphs $\{P(m,n)\}_{m\ge 2}$ is an expander family; equivalently, there exists a constant $c_n>0$ suc…

Is it true that for every positive rational number $r$ there exists a finite group $G$ such that $|\operatorname{Aut}(G)|/|G|=r$?

For every finitely generated group $G$, $G$ has co-context-free word problem if and only if $G$ embeds into Thompson's group $V$. Explicitly, for a finite generating set $X$ of $G$…

For every finite nonabelian group $G$, let $\exp(G)$ be the exponent of $G$. Define $s(G)$ to be the least integer $\ell$ such that every sequence over $G$ of length at least…

Every locally nilpotent countably categorical group is nilpotent; equivalently, for every group $G$, if $G$ is locally nilpotent and countably categorical, then $G$ is nilpotent.

In the mathematical theory of Kleinian groups, the density conjecture of Lipman Bers, Dennis Sullivan, and William Thurston, later proved independently by Namazi & Souto (2012) and…

For a field $K$ and a finite group $G$, say that $G$ is realizable over $K$ if there exists a Galois extension $L/K$ with $\operatorname{Gal}(L/K) \cong G$. Every finite group $G$…

Polylogarithmic-party hardness conjecture. There is no protocol for the $k$-party $t$-tuple interleaved group product over $G$ with $k=\log^c t$ parties and communication…

Define $f(N)$ be the minimal $k$ such that the following holds: if $G$ is an abelian group of size $N$ and $A\subseteq G$ is a random set of size $k$ then, with probability…

For every group $G$, let its non-commuting graph have vertex set $G$, with $g$ adjacent to $h$ exactly when $gh\ne hg$. If every clique of this graph is finite, does there exist a…