Dimension of elementary amenable groups

TL;DR

This paper investigates the homological and cohomological dimensions of elementary amenable groups, proposing a conjecture relating these dimensions to Hirsch length and providing proofs for specific classes and structural insights.

Contribution

It proves the conjecture for certain classes of groups, characterizes elementary amenable groups of homological dimension one, and explores cohomological dimension in the nilpotent-by-polycyclic case.

Findings

Proved the conjecture for abelian-by-polycyclic groups.

Elementary amenable groups of homological dimension one are filtered colimits of groups with cohomological dimension one.

Provided a detailed study of cohomological dimension for nilpotent-by-polycyclic groups.

Abstract

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.