46 problems

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…

Weak Andrews-Curtis conjecture. Every balanced presentation of the trivial group is $Q^{**}$-equivalent to

Basis-independent principal-overgroup conjecture.

Let ${\bf B}_n$ be the braid group with standard generators $\sigma_1,\ldots,\sigma_{n-1}$, and set $T_i=\sigma_i^2$. Let $G$ be the subgroup of ${\bf B}_n$ generated by…

Let $F_k$ be the free group of rank $k$. For a group $G$ and a normal subgroup $N\lhd G$, let $V_k(G,N)$ be the set of $k$-tuples that generate $N$ as a normal subgroup of $G$, and…

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…

Stable primitivity-rank conjecture. For every finitely generated subgroup $H\leq\mathbf{F}$,

For $n\geq 2$, consider the balanced presentation … Its relators are $(x^ny^{-n-1},xyxy^{-1}x^{-1}y^{-1})$, and its generators are $(x,y)$. Akbulut–Kirby conjecture. The relator pa…

Let $d\geq 3$ and $n\geq 3$ be integers. Consider a regular $d$-polytope of type $\{n,n,\dots,n\}$ with generating involutions $\rho_0,\rho_1,\dots,\rho_{d-1}$ and automorphism gro…

Let $q\in\mathbb{Q}\cap(0,4)$, and let … with $\Delta_q=\langle A,Q_q\rangle$. Kim–Koberda's non-freeness conjecture. The group $\Delta_q$ is not free for every…

Let $\mathbb{\mathbf{F}}$ be the free group, let $\mathrm{s\pi}(w)$ denote stable primitivity rank, and let $\pi(w)$ denote primitivity rank. Wilton's stability conjecture. For eve…

Let $\mathbb{\mathbf{F}}$ be the free group. Let $\mathrm{s\pi}(w)$ be the stable primitivity rank and let $\widetilde{\mathrm{s\pi}}(w)$ be the variant defined by restricting the…

Let $G$ be a finite group and let $A$ be a subset of $G$. A right sub-factor of $G$ related to $A$ is an object obtained from the right sub-factor construction described by the pre…

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…

A compact $2$-complex $X$ has nonpositive immersions if every compact, connected, collapsed combinatorial immersion $Y\rightarrow X$ has either trivial fundamental group or satisfi…

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…

Let $A$ be an alphabet, let $\overline{A}=A\cup A^{-1}$, and let $r$ be a reduced word in the free group on $A$. A prefix of $r$ is an initial segment of $r$ (and similarly for…

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

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

Reduction conjecture. The word

Even-endpoint conjecture. Faux CFC elements with full support only occur when the Coxeter graph has at least one even endpoint.

Let $u$, $v$, and $w$ be words, and let $u'$, $v'$, and $w'$ be cyclic permutations of $u$, $v$, and $w$, respectively. Let $f$ and $g$ be cyclic permutations of $v' * w'$ and…

Let $K_{\mathcal{P}}$ and $K_{\mathcal{Q}}$ be the standard $2$-dimensional CW-complexes of two finite presentations. Simple homotopy equivalence is generated by collapses and expa…

Polynomial-time conjugacy conjecture. The conjugacy problem can be solved in polynomial time for all hyperbolic elements of $H$.

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…