22 problems

The $G_{34}$ braid-group conjecture. The braid group associated with $G_{34}$ admits the presentation

The $G_{33}$ braid-group conjecture. The braid group associated with $G_{33}$ admits the presentation

The $G_{31}$ braid-group conjecture. The braid group associated with $G_{31}$ admits the presentation

The $G_{29}$ braid-group conjecture. The braid group associated with $G_{29}$ admits the presentation

Presentation conjecture. The homomorphism $\varphi_n:PL_n\to P_n^2$ is an isomorphism.

Let … be a graph-layer word in the twists $T_{ij}$, and let $\mathbf m=(m_1,\ldots,m_g)\in\mathbb Z^g$. Put … The associated framed graph-layer presentation is … A framed graph-lay…

Let $F_k$ be a free group of rank $k$, and let $G$ be a group. A subgroup $N\leq F_k$ with $F_k/N\cong G$ is a $G$-defining subgroup; write $\Sigma_k(G)$ for the set of all such su…

Stabilizer conjecture. The stabilizer of $W_l'$ is always either

Let $D_n$ be the dihedral group, with order-$n$ cyclic subgroup generated by the usual rotation. Let $S$ be a generating set composed of three involutions, none of which belongs to…

Almost-all non-congruence presentations conjecture. As the order of $S$ tends to infinity, the proportion of non-congruence presentations among all presentations tends to $1$:

Cocycle-Cheeger conjecture. The cocycle Cheeger constants in permutations of the Steinberg presentations have a uniform positive lower bound, at least when $n$ is fixed and…

Let $G$ be a group admitting a nice $(2,1)$-presentation $f\pi=\langle x,y\mid r\rangle$, and let $\mathcal{M}_\pi\subset H_1(G;\mathbb{R})$ be its associated marked polytope. Frie…

Let $G_{k+1}^k$ be one of the groups under consideration, and let $H_k$ be its subgroup generated by $b_1,\dots,b_k$. For distinct indices $i,j,m,l\in\{1,\dots,k\}$, consider the r…

Let $G$ be a group with $4$-periodic cohomology. Balanced-presentation conjecture. Then $G$ has a balanced presentation. This would imply that groups with $4$-periodic cohomology a…

Stable Andrews–Curtis conjecture. Every balanced group presentation for the trivial group may be changed to the trivial presentation by a finite sequence of stable Andrews–Curtis t…

Consistency conjecture. If

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…

Ivanov's asphericity conjecture. Then $\mathcal{Q}$ is aspherical.

Let $G$ be a group admitting a nice $(2,1)$-presentation $\pi=\langle x,y\mid r\rangle$, and let $\mathcal{M}_\pi\subset H_1(G;\mathbb{R})$ be its marked polytope. Two subsets are…

A planar presentation is a group presentation satisfying the paper's planar condition, namely admitting a suitable assignment of spin flags so that relations do not cross. Planar p…

Let $\Sigma_g$ be a closed surface of genus $g$, let $M_g$ be its mapping class group, and let $M_g[D]$ be the subgroup generated by the $D$-th powers of Dehn twists about all simp…

Let $S$ be a finite simple group and let $\tilde{S}$ be its universal cover. A finite group is proficient when it has the relevant profinite presentation property discussed in the…