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.



Kropholler's Conjecture and 3-manifolds
The paper of Scott and Swarup mentioned in the question cites Kropholler and Roller's 'Splittings of Poincaré duality groups'. On page 35 they write:



> 
> 'Let $K$ be a Poincaré duality group of dimension $(n−1)$ which is a subgroup of a Poincaré duality group $G$ of dimension $n$ ... Kropholler and Roller defined an obstruction $sing(K)$ to splitting $G$ over a subgroup commensurable with $K$ ... they showed that $sing(K)$ vanishes if and only if there is a $K$–almost invariant subset $Y$ of $G$ such that $KYK = Y$.'
> 
> 
> 


Taking n=3, this seems to answer your question.


