48 problems

Let $F_n=F(x_1,\dots,x_n)$ be the free group on generators $x_1,\dots,x_n$, and let $\langle x_1,\dots,x_n\mid r_1,\dots,r_n\rangle$ be a balanced presentation, with Andrews–Curtis…

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…

A balanced presentation of a group is a finite presentation with the same number of generators and relators. Generalized Andrews–Curtis conjecture. Any balanced presentation of the…

Stable Andrews–Curtis conjecture. Stable Andrews–Curtis moves should suffice to reduce every balanced presentation of the trivial group to the standard one-generator presentation.

Let $p$ be a prime. The coclass of a finite $p$-group is $n-c$, where $p^n$ is the order of the group and $c$ is the length of its lower central series. A coclass family is one of…

Let $n$ be a positive integer. Every finite simple group of order $n$ has a profinite presentation with $2$ generators and at most $2\lambda(n)$ relations, where, if…

Let $W(Y_{555})$ be the Coxeter group associated with the Coxeter diagram $Y_{555}$, whose generators are the reflections corresponding to the nodes of the diagram. Conway's bimons…

Let $G$ be a finite simple group, and let $\widetilde G$ denote its universal central extension. Wilson's conjecture. The group $\widetilde G$ has a presentation with $2$ generator…

Let $G$ be a finite simple group, and let a presentation of $G$ have a number of relations measured independently of the choice of generators. Mann's conjecture. Every finite simpl…

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

Andrews-Curtis conjecture. Two presentations with the same difference between the number of relators and the number of generators that are equivalent through moves (i)–(vii) are ac…

Projective presentation conjecture. The group $QL_n^2$ is isomorphic to $Q_n^2$.

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…

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…

Let $W$ be a complex reflection group, with complex braid group $\mathcal{B}_{\mathbb{C}}(W)$ and a BMR presentation for $W$. Exclude the groups $G_{24}$, $G_{27}$, $G_{29}$,…

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 $1$:

The DTF0L–EDT0L marked-group separation conjecture.

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…

Neutsch and Meyer's conjecture. The group $G$ is the Lyons group itself.

Let $M$ be a Garside monoid with two atoms, and let $G(M)$ denote its enveloping group. Picantin's conjecture. The group $G(M)$ is isomorphic to either a group of the form … for po…