480 problems

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…

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…

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…

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.

Let $G$ be a finitely generated group, and define its Schreier girth by … where $S$ is a minimal generating set and $S\sim_{\mathcal{S}}S'$ means that $S'$ can be obtained from $S$…

Let $G\subseteq GL(n,\mathbb{C})$ be a finitely generated subgroup. An element of $G$ is primitive if it belongs to a generating set obtainable from the fixed generating set by fin…

Non-factorability conjecture. On a large class of groups, not every SI SFT is an SFT factor of a contractible SFT.

A group is simple if it has no nontrivial proper normal subgroups, and finitely presented if it admits a presentation with finitely many generators and finitely many defining relat…

Let $f$ be the function on finite groups studied in the paper, with $f(G)=1$ precisely for cyclic groups. Second-smallest-value conjecture. $2$ is the second smallest value of the…

Let $L$ be a flag complex, let $\varphi\colon A_L\to\mathbb{Z}$ be an epimorphism, and let $BB_L^\varphi$ be the associated Artin kernel. Suppose $BB_L^\varphi$ is of type…

Let $\mathbb{F}$ be a skew field. Let $G$ be a torsion-free residually finite group of type $\mathsf{FP}_{n+1}(\mathbb F)$ such that the Hughes-free division ring…

Let $\varphi=\frac{1+\sqrt{5}}{2}$ be the golden ratio, and let $\langle\varphi\rangle=\{k\varphi:k\in\mathbb{Z}\}$ be the subgroup of $\mathbb{C}$ generated by $\varphi$. An irred…

Let ${\cal M}_g$ be the mapping class group. Recall that a group ${\varGamma}$ is uniformly perfect if there is a natural number $N$ such that every element of ${\varGamma}$ is a p…

Let ${\cal LF}_n$ be a locally free group, and consider its random walk with word length $K$ and drift $l$, the limiting expected normalized word length. Let $T(w)$ be the roof of…

Let ${\cal LF}_n^+$ be the locally free semigroup (the heap) and ${\cal LF}_n$ the locally free group (the colored heap). For a random walk word $w_N$, let $T(w_N)$ denote its roof…

Let $G$ be an infinite simple tame $K^*$-group, where a $K^*$-group is a group of finite Morley rank in which every proper definable subgroup is a $K$-group, and a $K$-group is one…

Let $G(\Gamma)$ be the group associated with the LOT $\Gamma$, let $V(\Gamma)$ be its vertex set, and let $B$ be the subgroup generated by the alternating words in $V(\Gamma)$. A D…

Let $N$ be a compact, connected, ${\mathbf{P}^2}$-irreducible $3$-manifold whose boundary is a non-empty collection of incompressible tori and Klein bottles. Assume that every subg…

Let $M$ be a closed, connected, ${\mathbf{P}^2}$-irreducible $3$-manifold with infinite fundamental group. Rational subgroup conjecture. Every subgroup of $\pi_1(M)$ which embeds i…

Let $M$ be a closed, connected, ${\mathbf{P}^2}$-irreducible $3$-manifold with infinite fundamental group. For a subgroup $H\leq\pi_1(M)$, write … for its normalizer. Normalizer Co…

Let $V$ be the ground model. For a centreless group $G\in V$, let $\tau(G)$ be the least ordinal $\alpha$ such that the automorphism tower satisfies $G_\beta=G_\alpha$ for all…

Let $\Gamma$ be an infinite group with Kazhdan property (T), and let $(\Gamma_n)$ be a normal chain in $\Gamma$ with trivial intersection. Write $\mathrm{RG}(\Gamma,(\Gamma_n))$ fo…

Let $n$ be the order of a Moufang loop, and let $\operatorname{Mlt}(L)$ denote its multiplication group. Polynomial-bound conjecture. The function … solves the problem of finding a…

Let $\Gamma\subset \operatorname{Mob}(\mathbb S^n)$ be an arithmetic lattice with $n\ge 3$. Kapovich–Potyagailo–Vinberg's conjecture. The group $\Gamma$ is noncoherent: it contains…