**Martin R. Bridson and Peter H. Kropholler**

*To appear in Crelle's Journal. Submitted 2 Sept 2012, accepted 29 Nov 2012.*

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.

31 pages, 2 figures.