7 problems

A subgroup $A\subset \pi_1(N)$ is separable if for every $g\in\pi_1(N)\setminus A$ there is an epimorphism $\alpha:\pi_1(N)\to G$ to a finite group such that…

Let $A=A(\Gamma)$ be an Artin group, let $\mathbb{S}$ be the set of full subgraphs $\Sigma$ of $V(\Gamma)$ such that $A(\Sigma)$ is an irreducible spherical Artin group, and let…

Profinite closure conjecture. Every such subset is closed in the profinite topology.

Aschenbrenner–Friedl–Wilton's conjecture. If $H$ is a fully relatively quasiconvex subgroup of $\pi_1M$, then $H$ is a virtual retract. In particular, $H$ is separable.

Let $N$ be an orientable, compact, non-positively curved $3$-manifold with empty or toroidal boundary, and write $\pi=\pi_1(N)$. Equip $\pi$ with its natural relatively hyperbolic…

Let $N$ be a compact, orientable, irreducible $3$-manifold with empty or toroidal boundary. Suppose no torus of the JSJ decomposition bounds a Seifert fibered $3$-manifold on both…

LERF conjecture. The group $\pi_1(N)$ is LERF; equivalently, every finitely generated subgroup of $\pi_1(N)$ is separable in $\pi_1(N)$.