Dimension of elementary amenable groups

Abstract

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 hd k (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.