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? Author :zeraoulia rafik
I would like someone to give me some explanation (link, paper,..) about this strange problem. Thank you for your replies or any comments.
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.