Groups possessing extensive hierarchical decompositions

TL;DR

This paper explores the hierarchical structure of Kropholler's class of groups, constructing examples at various stages of the hierarchy, including groups with torsion and torsion-free groups at specific levels.

Contribution

It demonstrates the existence of countable groups at each hierarchical stage of Kropholler's class, extending previous knowledge limited to early stages, and includes groups with torsion.

Findings

Countable groups exist at each stage of the hierarchy.

Construction of torsion groups at higher stages.

Existence of torsion-free groups at the third stage.

Abstract

Kropholler's class 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 the class, then $G$ is itself in the class. Kropholler's class admits a hierarchical structure, i.e., a natural filtration indexed by the ordinals. For example, stage 0 of the hierarchy is the class of all finite groups, and stage 1 contains all groups of finite virtual cohomological dimension. We show that for each countable ordinal $\alpha$, there is a countable group that is in Kropholler's class which does not appear until the $\alpha+1$st stage of the hierarchy. Previously this was known only for $\alpha= 0$, 1 and 2. The groups that we construct contain torsion. We also review the construction of a torsion-free group that lies in the third stage of the hierarchy.