15 problems

Property H conjecture. Every Artin–Tits presentation satisfies Property $H$. The conjecture asks whether transformations of type $\infty$ are unnecessary for reducing every word re…

Fix $k\ge 2$ and $m\ge 1$, and let $F=F(a_1,\dots,a_k)$ be the free group of rank $k$. For an $m$-tuple $\sigma$ of elements of $F$, write $R(\sigma)$ for the corresponding set of…

The perfect Prishchepov group type conjecture. If $P(r,n,k,r-1,q)$ is perfect, then it is of type $\widetilde{\mathfrak{Z}}$. This conjecture predicts that every perfect nontrivial…

A compact $2$-complex $X$ has negative immersions when it satisfies the negative-immersion property defined in the source. An immersion is a map $\psi:H\rightarrow F$ between graph…

Let $P_2(w)$ be a non-orientable cyclic presentation and let $G_2(w)$ be its associated group. Free-subgroup conjecture. Either $G_2(w)$ contains a non-abelian free subgroup, or…

Pure twin group conjecture. $PT_n$ is a right-angled Artin group for each $n\geq 3$.

Let $S^{TFFF}(n)$ and $S^{FFFF}(n)$ be the classes of cyclically presented groups defined in the source, and let $\Gamma_n(1,n/2-1)$ be the corresponding group. Exclusion conjectur…

Let $G$ be a finitely generated group of rank $n$. An ordered $n$-tuple is normally generating when its components normally generate $G$, and its recalcitrance is the least number…

Let $G$ be a finitely generated group, let $w(G)$ be its weight, and let $G_{ab}=G/[G,G]$ be its abelianization. For $n\geq\max(w(G),2)$, an ordered $n$-tuple is normally generatin…

Let $F_2$ be the free group on $x,y$, let ${\mathcal B}_2$ be the set of balanced presentations of the trivial group with two generators, and let $\operatorname{Aut}(F_2)$ act comp…

Let $\mathcal{K}$ be a directed $2$-complex, let $u$ be a $1$-path, and let $H$ be a subgroup of the diagram group $\mathrm{DG}(\mathcal{K},u)$. A subgroup is closed for components…

Let $L$ be a finite, contractible 2-complex. A 3-deformation is the operation, described in the source, of successively gluing 3-balls along open 2-cells and performing inverse ope…

Let $\mathcal{L}$ be a line arrangement, and let $G$ be a group. They are conjugation-free when their defining relations have no conjugations; they are decomposable when their asso…

Let $\Gamma$ be a planar graph, let $B_n\Gamma$ be its braid group, and let $P_3\Gamma$ be its pure braid group on three points. A group is simple-commutator-related if it has a pr…

Planar graph braid group commutator presentation conjecture. There exists a presentation for $B_2\Gamma$ such that the relators are all commutators corresponding to pairs of disjoi…