7 problems

Let $G$ be a finitely generated group and let $H$ be a subgroup. An $H$-almost invariant subset $A$ is $H$-proper if it is not equivalent to either the empty set or $G$ modulo fini…

Let $k$ be a positive integer and write $S(k)=\{1,2,\dots,k\}$. Let $G$ be a finite abelian group. Say that $S(k)$ splits $G$ if there is a subset $T\subseteq G$ such that the tran…

Swarup's Strong Accessibility Conjecture. Decompose $G$ maximally over finite subgroups, then decompose each resulting vertex group maximally over two-ended subgroups, and repeat t…

Group-pair splitting conjecture. There is an analogue for group pairs of the preceding theorem on $it{PD}^3$ groups.

Acylindrical vertex-pair conjecture. Either $s_i=1$ (and hence $s_j=1$), or $i=j$ and $g^{-1}H_rg=H_r$.

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

Jankiewicz's splitting conjecture. The Artin group $\mathrm{Art}_{23M}$ splits as a graph of finite-rank free groups.