Decomposition theorems for group pairs

Muller, H.

doi:10.1007/bf01261870

Cited by 15 publications

(9 citation statements)

References 12 publications

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“…Actually, if one wants to prove the Torus Theorem alone, it is possible to dispense with many of the technicalities of this paper. This is partly because one only needs group-theoretic results about PD 3 -groups, and partly because the use of modest geometric methods avoids the need for the difficult group-theoretic techniques of Miiller [25]. The goal of [20] is a complete proof of the Torus Theorem in this spirit.…”

Section: The Torus Theorem and ?D 3 -Groupsmentioning

confidence: 99%

An Analogue of the Torus Decomposition Theorem for Certain Poincaré Duality Groups

Kropholler

1990

Proceedings of the London Mathematical Society

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It is shown that Poincare* duality groups which satisfy the maximal condition on centralisers have a canonical decomposition as the fundamental group of a finite graph of groups in which the edge groups are polycyclic-by-finite. The results give useful information only when there are large polycyclic subgroups. Since 3-manifolds groups satisfy Max-c, the results provide a purely grouptheoretic proof of the Torus Decomposion Theorem. In general, fundamental groups of closed aspherical manifolds satisfy Poincare" duality and in fact many of the known examples satisfy Max-c. Thus the results provide a new approach to aspherical manifolds of higher dimensions.A. M.S. (1980) subject classification: 20E06.

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Section: The Torus Theorem and ?D 3 -Groupsmentioning

confidence: 99%

An Analogue of the Torus Decomposition Theorem for Certain Poincaré Duality Groups

Kropholler

1990

Proceedings of the London Mathematical Society

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“…Namely, n(S0)=2. Thus Corollary 3.3 of [8] can be applied; among the cases listed there only the nine types above can occur. To…”

Section: H~(t;ffg) --+H (G~s_~wt;2~g)mentioning

confidence: 96%

“…Proof. Since So+G, the "weight" n(So) with respect to the pair (G; $1 .... , S,,) is easily determined by (3.5) and Lemma 3.1 (for the definition and properties of weight see [8]). Namely, n(S0)=2.…”

Section: H~(t;ffg) --+H (G~s_~wt;2~g)mentioning

confidence: 98%

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Plane motion groups and virtual Poincar� duality of dimension two

Eckmann

Muller

1982

Invent Math

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“…This result will be used in a forthcoming paper by Guirardel, Scott and Swarup on relative versions of the algebraic torus theorem and other results. These results concern splittings "adapted" to a family of subgroups, a concept that was introduced by Müller in [17].…”

Section: Contentsmentioning

confidence: 99%

Splittings of non-finitely generated groups

Lassonde

2012

Algebr. Geom. Topol.

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In geometric group theory one uses group actions on spaces to gain information about groups. One natural space to use is the Cayley graph of a group. The Cayley graph arguments that one encounters tend to require local finiteness, and hence finite generation of the group. In this paper, I take the theory of intersections of splittings of finitely generated groups (as developed by Scott, Scott-Swarup, and Niblo-Sageev-Scott-Swarup), and rework it to remove finite generation assumptions. Whereas the aforementioned authors relied on the local finiteness of the Cayley graph, I capitalize on the Bass-Serre trees for the splittings.

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Decomposition theorems for group pairs

Cited by 15 publications

References 12 publications

An Analogue of the Torus Decomposition Theorem for Certain Poincaré Duality Groups

An Analogue of the Torus Decomposition Theorem for Certain Poincaré Duality Groups

Plane motion groups and virtual Poincar� duality of dimension two

Splittings of non-finitely generated groups

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