Januszkiewicz, T., Kropholler, P.H. and Leary, I.J.
Groups possessing extensive hierarchical decompositions
Bulletin of the London Mathematical Society, 42, (5), . (doi:10.1112/blms/bdq045).
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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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