Abstract
This paper has three parts. It is conjectured that for every elementary amenable group G and every non-zero commutative ring k, the homological dimension hdk(G) is equal to the Hirsch length h(G) whenever G has no k-torsion. In Part I this conjecture is proved for several classes, including the abelian-by-polycyclic groups. In Part II it is shown that the elementary amenable groups of homological dimension one are colimits of systems of groups of cohomological dimension one. In Part III the deep problem of calculating the cohomological dimension of elementary amenable groups is tackled with particular emphasis on the nilpotent-by-polycyclic case, where a complete answer is obtained over ℚ for countable groups.
Funding source: EPSRC
Award Identifier / Grant number: Senior Fellowship
Funding source: Royal Society
Award Identifier / Grant number: Wolfson Research Merit Award
Funding source: Cornell University
Award Identifier / Grant number: Visiting Professorship
We thank the referee for reading our original draft so carefully and for providing such constructive comments.
© 2015 by De Gruyter
