3 problems

Let $G$ be a finitely generated group hyperbolic relative to a finite collection $\{P_1,\ldots,P_n\}$ of proper finitely generated subgroups. Semistability conjecture. If each…

The splitting conjecture for finitely presented non-semistable groups. If $G$ does not have semistable fundamental group at infinity, then $G$ splits non-trivially.

The lamplighter group's non-semistability conjecture. The group $L$ is not semistable at infinity.