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An Analogue of the Torus Decomposition Theorem for Certain Poincaré Duality Groups
Author(s) -
Kropholler P. H.
Publication year - 1990
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-60.3.503
Subject(s) - mathematics , poincaré duality , torus , duality (order theory) , pure mathematics , group (periodic table) , graph , finite group , maximal torus , combinatorics , geometry , fundamental representation , lie algebra , chemistry , cohomology , organic chemistry , weight
It is shown that Poincaré 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 Poincaré 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.