This short note, produced for the volume of conjectures produced on the occasion of Guido Mislin's retirement, describes the status of the Kropholle conjecture which may be viewed as a far reaching generalisation of Stallings' theorem on groups with more than one end

Niblo, Graham A.

Sageev, Michah

English

Southampton (e-Prints Soton)

1. Graham A. Niblo and Michah Sageev: The KrophollerconjectureA ﬁnitely generated group G is said to split over subgroup H ifand only if G may be decomposed as an amalgamated free productG = A CB (with A  = C  = B) or as an HNN extension G = A C. TheKropholler conjecture is concerned with the existence of such splittings.Given a subgroup H of a ﬁnitely generated group G the invariante(G,H) is deﬁned to be the number of Freudenthal (topological) ends ofthe quotient of the Cayley graph of G under the action of the subgroupH. This number does not depend on the (ﬁnite) generating set chosenfor G [3] so it is an invariant of the pair (G,H). For example, if G is afree abelian group and H is an inﬁnite cyclic subgroup then e(G,H) =0, if G has rank 1, e(G,H) = 2 if G has rank 2 and e(G,H) = 1 if G hasrank greater than or equal to 3. This invariant generalises Stallings’deﬁnition of the number of ends of the group G since if H = {1} thene(G,H) = e(G).In [4] Stallings showed that the group G splits over some ﬁnite sub-group C if and only if e(G)   2. There are several important gener-alisations of this fact, the most wide ranging being the algebraic torustheorem, established by Dunwoody and Swenson [1]. This states that,under suitable additional hypotheses, if G contains a polycyclic-by-ﬁnite subgroup H of Hirsch length n with e(G,H)   2 then either(1) G is virtually polycyclic of Hirsch length n + 1,(2) G splits over a virtually polycyclic subgroup of Hirsch length n,(3) G is an extension of a virtually polycyclic group of Hirsch lengthn   1 by a Fuchsian group.This theorem generalises the classical torus theorem from low dimen-sional topology which asserts that a closed 3-manifold which admits animmersed incompressible torus either admits an embedded incompress-ible torus or has a Seifert ﬁbration. These topological conclusions implythe algebraic conclusions for the fundamental group of the manifold.An important ingredient of the proof of the algebraic torus theoremis a special case of the so called Kropholler conjecture. Its originalformulation relies on the following observation of Scott:A subgroup H of a ﬁnitely generated group G satisﬁes e(G,H)   2if and only if G admits a subset A satisfying the following:(1) A = HA,(2) A is H-almost invariant, and(3) A is H-proper, i. e. , neither A nor G   A is H-ﬁnite.1We will refer to the subset A as a proper H-almost invariant subset.In his proof of the algebraic torus theorem for Poincar´ e duality groupsKropholler observed that, under certain additional hypotheses, if Gadmits a proper H-almost invariant set A such that A = AH then Gadmits a splitting over some subgroup C < G related to H (see [2] foran outline of the proof). He conjectured that the additional hypotheseswere inessential. Speciﬁcally:Conjecture 1.1 (The Kropholler conjecture). Let G be a ﬁnitely gen-erated group and H < G. If G contains a proper H-almost invariantsubset A such that A = AH then G admits a non-trivial splitting overa subgroup C which is commensurable with a subgroup of H.The conjecture is known to hold when G is a Poincar´ e duality groupor when G is word hyperbolic and H is a quasi-convex subgroup. Ingeneral it is known (for an arbitrary ﬁnitely generated group G) when-ever H is a subgroup which satisﬁes the following descending chaincondition:Every descending chain of subgroups H = H0   H1   H2   ... suchthat Hi+1 has inﬁnite index in Hi eventually terminates.This condition holds for example for the class of ﬁnitely generatedpolycyclic groups, in which class the Hirsch length is the factor con-trolling the length of such a chain. This is a key ingredient in the proofof the full algebraic torus theorem.An alternative, more geometric, point of view on the conjecture isprovided by the following characterisation:Theorem 1.2. Given a ﬁnitely generated subgroup G and a subgroupH < G the invariant e(G,H) is greater than or equal to 2 if and onlyif G acts with no global ﬁxed point on a CAT(0) cubical complex withone orbit of hyperplanes, and so that H is a hyperplane stabiliser. Hadmits a right invariant, proper H-almost invariant subset if and onlyif the action can be chosen so that H has a ﬁxed point in the complex.References[1] M. J. Dunwoody and E. L. Swenson, ’The algebraic torus theorem’, Invent.Math. 140 (2000), no. 3, pp. 605–637.[2] P. H. Kropholler, A group theoretic proof of the Torus Theorem, in: (eds.G. A. Niblo and M. A. Roller) Geometric Group Theory, Sussex 1991, Volume1, (Cambridge University Press 1993), pp. 138158.[3] G. P. Scott, ‘Ends of pairs of groups’, Journal of Pure and Applied Alge-braVol. 11, 1977, pp 179–198.[4] J. R. Stallings, ‘On torsion-free groups with inﬁnitely many ends’, Ann. ofMath. 88, (1968), pp. 312–334.2

On the Kropholler conjecture

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