  The paper has three parts. It is conjectured that for every elementary
amenable group G and every non-zero commutative ring k, the homological
dimension of G over k is equal to the Hirsch length of G whenever G has no
k-torsion. In Part I this is proved for several classes, including the
abelian-by-polycyclic groups. In Part II it is shown that elementary amenable
groups of homological dimension one are filtered colimits of systems of groups
of cohomological dimension one. Part III is devoted to the deeper study of
cohomological dimension with particular emphasis on the nilpotent-by-polycyclic
case.
