Groups possessing extensive hierarchical decompositions
Groups possessing extensive hierarchical decompositions
The class HF is the smallest class of groups that contains all finite groups and is closed under the following operator: whenever G admits a finite-dimensional contractible G-CW-complex in which all stabilizers are in HF, then G is itself in HF. The class HF admits a natural filtration by the ordinals. For each countable ordinal we show that there is a countable group that is in HF but has not arisen by stage alpha of this filtration. Previously this result was known only for alpha equal to 0, 1, and 2. The groups that we construct contain torsion. We also review the construction of a torsion-free countable group in HF that is not in stage 2 of the filtration.
896-904
Januszkiewicz, T.
a3b164ea-3189-4ce2-b67c-7d0b23c4adb6
Kropholler, P.H.
0a2b4a66-9f0d-4c52-8541-3e4b2214b9f4
Leary, I.J.
57bd5c53-cd99-41f9-b02a-4a512d45150e
October 2010
Januszkiewicz, T.
a3b164ea-3189-4ce2-b67c-7d0b23c4adb6
Kropholler, P.H.
0a2b4a66-9f0d-4c52-8541-3e4b2214b9f4
Leary, I.J.
57bd5c53-cd99-41f9-b02a-4a512d45150e
Januszkiewicz, T., Kropholler, P.H. and Leary, I.J.
(2010)
Groups possessing extensive hierarchical decompositions.
Bulletin of the London Mathematical Society, 42 (5), .
(doi:10.1112/blms/bdq045).
Abstract
The class HF is the smallest class of groups that contains all finite groups and is closed under the following operator: whenever G admits a finite-dimensional contractible G-CW-complex in which all stabilizers are in HF, then G is itself in HF. The class HF admits a natural filtration by the ordinals. For each countable ordinal we show that there is a countable group that is in HF but has not arisen by stage alpha of this filtration. Previously this result was known only for alpha equal to 0, 1, and 2. The groups that we construct contain torsion. We also review the construction of a torsion-free countable group in HF that is not in stage 2 of the filtration.
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e-pub ahead of print date: 17 June 2010
Published date: October 2010
Organisations:
Pure Mathematics
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Local EPrints ID: 199421
URI: http://eprints.soton.ac.uk/id/eprint/199421
ISSN: 0024-6093
PURE UUID: 7dae0a77-d744-43d0-bacf-040564a89c0a
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Date deposited: 18 Oct 2011 11:01
Last modified: 15 Mar 2024 03:46
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Author:
T. Januszkiewicz
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