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Groups possessing extensive hierarchical decompositions

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The class HF of groups is the smallest class of groups which contains all finite groups and is closed under the following operator: whenever G admits a finite-dimensional contractible G-CW-complex in which all stabilizer groups are in HF, then G is itself in HF. The class HF admits a natural filtration indexed by the ordinals. For example, H0F is the class of all finite groups and H1F contains all groups of finite virtual cohomological dimension. We show that, for each countable ordinal a, there is a countable group that is in HF \ HaF. Previously this was known only for a = 0, 1 and 2. The groups that we construct contain torsion. We also review the construction of a torsion-free group that lies in HF \ H2F.

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