On groups of finite cohomological dimension

Abstract

The celebrated theorem that a group of cohomological dimension one is free was first proved by Stallings [8] under the assumption that the group is finitely generated. Later Swan [lo] was able to obtain this theorem without such an assumption. Thus we have that the cohomological dimension (cd) of a group G is the same as its category and geometric dimension with the only possible exception when cd G is two (2). Groups of cd 2 are by no means rare [3, Section 8.31. This paper stems from an attempt to classify such groups. We find that the simple infinite groups constructed by Camm [l] are of cd 2 (Section 4), and a locally nilpotent group of cd 2 turns out to be Z @ Z or a noncyclic subgroup of Q (Section 3, Theorem C). Recent work of Gruenberg [3] and Stammbach [9] on locally nilpotent groups and solvable groups, relate the cd of such groups to their Hirsch numbers. We obtain that a torsion-free solvable group is of finite cd if and only if it has finite Hirsch number (Section 2, Theorem I). With this grouptheoretical interpretation of cd we shall prove that a locally nilpotent group is of finite cd if and only if its abelian subgroups are of finite cd. (Theorem A). Moreover, in this case, it is merely a subgroup of some unitriangular group T&J of various degrees d (Theorem B). The Mal’cev completion G* of a torsion-free locally nilpotent group G is used as the basic tool throughout this paper.