Suppose that $G$ is a group and that $H$ is a subgroup, both finitely generated, and assume that there is a non-trivial H-almost invariant set $X$ with $HXH=X$. Kropholler's Conjecture asserts that $ G $ splits over a subgroup commensurable with a subgroup of $H$.
Note that the algebraic hypothesis $HXH= X $ can be reformulated in terms of strong crossings - see "Splittings of Groups and Intersection Numbers" by Peter Scott and Gadde A. Swarup.
There is a paper of M.Sageev which proves the conjecture for quasiconvex subgroups of hyperbolic groups. My question is this: is the result true for $3$-manifold groups and surface subgroups?
I would like someone to give me some explanation (link, paper,..) about this strange problem. Thank you for your replies or any comments.