Comments on: Open Problems
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Recent Progress and Open ProblemsMon, 26 Mar 2012 00:38:54 +0000
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By: gaddeswarup
https://ldtopology.wordpress.com/open-problems/#comment-1450
Sun, 17 Apr 2011 23:55:05 +0000http://ldtopology.wordpress.com/open-problems/#comment-1450Here is a problem due to Kropholler and Roller which is bothering me in retirement. It is possibly not very interesting for many but a special case of it turned out be useful and is one half of the algebraic torus theorem and was proved by Dunwoody and Roller in 1986. The question is this: Suppose G is group and H a subgroup, both finitely generated and assume that there is a non-trivial H-almost invariant set X with HXH=X. Then the ‘conjecture’ is that G splits over a subgroup commensurable with a subgroup of H. The algebraic hypothesis HXH=X can be reformulated in terms of strong crossings (done by Peter Scott and myself in the paper “Splittings of Groups and Intersection Numbers”). Apart from the result of Dunwoody Roller in the virtually polycyclic case (the proofs actually give more), there is a papaer of M.Sageev which proves the conjecture for quasiconvex subgroups of hyperbolic groups. I think that the result is true for 3-manifold groups and surface subgroups. It is a strange problem whose firther uses are not clear but it keeps bugging me.
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