Authors: Tadeusz Januszkiewicz, Peter H Kropholler and Ian J Leary

Tadeusz Januszkiewicz
Affiliation: The Ohio State University and the Polish Academy of Sciences 

Peter Kropholler
Affiliation: University of Glasgow 

Ian Leary
Affiliation: The Ohio State University and Heilbronn Institute
University of Bristol 

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.  

Status: preprint