44 problems

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

Let $G$ be a one-relator group, and let $BS(1,n)=\langle a,t\mid ta t^{-1}=a^n\rangle$ denote the Baumslag–Solitar group for an integer $n\neq 0$. Gersten's conjecture. If $G$ cont…

Let $G$ be a hyperbolic one-relator group. Wise's conjecture. Every such group is virtually free-by-cyclic. If true, this would imply that every hyperbolic one-relator group admits…

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.

Baumslag's conjectures.

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…

A strictly ascending HNN extension of a finite rank free group is an HNN extension $F_k*_{\theta}$, where $\theta:F_k\rightarrow F_k$ is injective but not surjective. The groups un…

Let … A one-relator group is a group admitting a presentation with one defining relator. Gersten's conjecture. The group $G_{(1,2)}$ has the highest Dehn function among all one-rel…

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…

Let $G$ be the Baumslag-Gersten one-relator group … For a finitely generated group, the conjugator length function measures the smallest length of a conjugator between two conjugat…

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…

Baumslag–Wise conjecture. Every one-relator group with torsion is virtually free-by-cyclic, and every hyperbolic one-relator group is virtually free-by-cyclic.

A group is coherent if every finitely generated subgroup is finitely presented. A one-relator group is a group admitting a presentation with one defining relator. Baumslag's conjec…

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}…

Let $G$ be a nontrivial group. Adding one generator and then one relator means forming a quotient of $G\star\mathbb Z$ by the normal closure of an element. Kervaire's conjecture. T…

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 $n$.

Let $M=\operatorname{Inv}\langle A\mid w=1\rangle$ be a special one-relator inverse monoid, with defining word $w$. Decompose $w$ into its minimal invertible pieces using the Benoi…