24 problems

Geometric presentation conjecture. The group $\langle x_1,\ldots,x_n\mid w\rangle$ is the fundamental group of an orientable 3-manifold if and only if $w$ is a geometric word.

Let $\mathcal{G}'(r)$ denote the random model of two-generator one-relator groups whose defining relation is a commutator, and let $p_r$ be the probability that a group…

Hyperbolicity conjecture. Every such group $G$ is hyperbolic.

The torsion-free cyclic relation-module conjecture. If $G$ is torsion-free and $N_{\mathbb{Q}}$ is cyclic as a $\mathbb{Q}G$-module, then $N$ is normally generated by a single elem…

Let $F$ be a free group, let $w\in F$, and let $G=F/\langle\!\langle w\rangle\!\rangle$ be the corresponding one-relator group. For a subset of $F$, write…

Mel'nikov's conjecture. An infinite and residually finite Mel'nikov group is either a free group, a surface group or a solvable Baumslag--Solitar group.

Let $\{A_\lambda\}$ be a family of groups and let $\star_\lambda A_\lambda$ be their free product. For a fixed factor $A_\lambda$, let $w$ be an element of the free product that is…

Let $A$ be a nontrivial group, set $H=A\star\mathbb Z$, and let \savebox{\@brx}{m@th{langle}}% mathopen{copy@brxkern-0.5wd@brxusebox{@brx}} wsavebox{@brx}{\m@th{\rangle}…

Let $A$ be a group, let $H=A\star\mathbb Z$, and let $p:H\to\mathbb Z$ be the standard projection to the $\mathbb Z$ factor. For $w\in H$, write \savebox{\@brx}{m@th{langle}…

Let $A$ and $B$ be torsion-free groups, set $H=A\star B$, and let \savebox{\@brx}{m@th{langle}}% mathopen{copy@brxkern-0.5wd@brxusebox{@brx}} wsavebox{@brx}{\m@th{\rangle}…

For each integer $k\geq 2$, let … The subgroup membership conjecture for $B(P_{3,k})$. The subgroup membership problem is decidable in $B(P_{3,k})$ for every $k\geq 2$. The subgrou…

Louder--Wilton conjecture. Every one-relator group whose presentation complex has negative immersions is hyperbolic.

Infinite-index subgroup surface-group conjecture. Then $G$ is a surface group.

Surface Group Conjecture A. Let $G$ be a residually finite Mel'nikov group. Then $G$ is a surface group or $\operatorname{BS}(1,n)$ for some non-zero integer $n$.

Weak Gersten conjecture. Every one-relator RG group is hyperbolic.

Baumslag's conjectures.

Positive-theory conjecture. A non-cyclic one-relator group has trivial positive theory unless it is isomorphic to $BS(1,n)$ for some $n\in\mathbb Z$. This is the positive-theory an…

Let $F$ be a free group, let $w\in F\setminus\{1\}$ have primitivity rank $\pi(w)=2$, and let $G=F/\langle\!\langle w\rangle\!\rangle$. Let $P<G$ be the peripheral subgroup supplie…

Let $G$ be a one-relator group whose presentation complex has negative immersions. One-relator hyperbolization conjecture. The group $G$ is hyperbolic. The conjecture would imply t…

Wise's conjecture. Every torsion-free one-relator group with no subgroup isomorphic to $BS(m,n)$, for $m\neq\pm n$, acts freely on a CAT(0) cube complex.

One-relator action conjecture. Some nontrivial one-relator group acts translation-like on $G$.

Let $G$ be a one-relator group with torsion that does not admit a free group as a free factor. Since one-relator groups with torsion are hyperbolic, $G$ has a JSJ-decomposition, me…

A one-relator group with torsion is a group with a presentation $G=\langle X;R^n\rangle$, where $R$ is not a proper power and $n>1$. Pride's folk conjecture. Apart from the complic…

Let $G$ be a finitely generated one-relator group. Theorem 1.1 establishes proper $3$-realizability under the hypothesis that $G$ has finitely many ends. Proper 3-realizability con…