Kropholler–Roller conjecture on splittings from codimension-1 subgroups

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 finitely many $H$-cosets. Kropholler–Roller conjecture. If $G$ contains an $H$-proper $H$-almost invariant subset $A$ such that $HAH=A$, then $G$ admits a nontrivial splitting over a subgroup commensurable with a subgroup of $H$. This strengthens Kropholler's formulation by requiring both left and right $H$-invariance. It is presented as open in the source; the paper discusses results under Poincaré duality, virtual polycyclicity, and commensuration hypotheses.