13 problems

A group is isolated if it is an isolated point in the space of marked groups, and a subgroup is non-isolated if it is not isolated there. Finite-index subgroup conjecture. There ex…

Let $G$ be an amenable finitely generated group. Say that $G$ can be approximated by elementary amenable groups when it is a limit of such groups in the space of marked groups. Ste…

The DTF0L–EDT0L marked-group separation conjecture.

LEF non-co-semi-decidability conjecture. The set of LEF groups is not $\Lambda_{WP}$-co-semi-decidable.

Finite-presentation isolated-groups conjecture. The set of isolated groups is not $\Lambda_{FP}$-semi-decidable.

Finite-recognizability conjecture. Finite groups are the only abstractly recognizable groups in $(\mathcal{G}_{WP},d,\nu_{WP})$.

Solvable-word-problem preformation conjecture. There exists a group $H$, non-isomorphic to $G$, with solvable word problem and rank at most $k+1$, such that $G$ preforms $H$.

Marked-isolated-groups conjecture. Marked isolated groups are the only marked groups that can be recognized for $\Lambda_{WP}$; equivalently, they are the only marked groups whose…

LEF and isolated-groups conjecture. The set of LEF groups is not $\Lambda_{WP}$-co-semi-decidable, and the set of isolated groups is not $\Lambda_{WP}$-semi-decidable.

Decidability–clopenness conjecture. Every $\Lambda_{WP}$-decidable property of $\mathcal{G}_{WP}$ is clopen.

Main conjecture. Any $\Lambda_{WP}$-computable function defined on $\mathcal{G}_{WP}$, with image in any recursive metric space, is continuous.

Let $F$ be a finitely generated free group, let $M_k,M$ be normal subgroups of $F$, and identify $\operatorname{MG}(F)$ with the space of marked groups. Suppose that $M_k$ converge…

Torsion-free marked groups conjecture. For every $k\geq 3$, there are such marked groups $\Gamma_k$ and $\Delta_k$ with each group torsion free.