9 problems

Let $G$ be a finitely presented group, and let $H^2(G;\mathbb{Z}G)$ denote its second cohomology group with coefficients in the group ring $\mathbb{Z}G$. Freeness conjecture. The a…

Let $A$ be a countable abelian group, and write $\bigoplus_\omega(\mathbb{Q}\oplus\mathbb{Q}/\mathbb{Z})$ for the countable direct sum of copies of…

A Hopfian finitely presented group is a finitely presented group for which every surjective endomorphism is an automorphism; its word problem asks whether a given word represents t…

Finite presentability conjecture. Then $H$ is finitely presented.

The groups under consideration are finitely generated simple Kac–Moody groups whose Weyl groups have Coxeter diagrams containing entries equal to $\infty$. Abramenko's conjecture.…

Let $G$ be a finitely presented group satisfying the condition that there is a finite set of virtually cyclic subgroups $\{H_1,\ldots,H_n\}$ such that every virtually cyclic subgro…

Baumslag–Pride conjecture. Every deficiency one presentation with a proper power relator should define a large group.

Let $G$ be a finitely presented group. A free subsemigroup on two generators is a subsemigroup generated by two elements with no nontrivial relations. Grigorchuk's stronger conject…

Embeddability-complexity conjecture. The set $K$ is $\Sigma^0_3$-complete. Equivalently, deciding whether $\overline{P}_i$ embeds into $\overline{P}_j$ for finite presentations…