Questions on Lie Algebras of Cohomological Dimension 1

A Lie algebra L is said to be of cohomological dimension 1, if H(L, M) = 0 for any L-module M . Due to the standard interpretation of the second cohomology group, this is equivalent to the condition that each exact sequence 0 → M → G → L → 0 splits. Of course, the cohomological dimension may be defined via standard devices of homological algebra – e.g. as the minimal length of the projective resolution. Particulary, the latter definition shows, that for Lie algebras of cohomological dimension 1 all higher cohomology groups also vanish. The similar notion may be defined for other classes of algebraic systems, e.g. for groups. Note that we consider the category of all L-modules, including infinite-dimensional ones. If we restrict ourselves with, say, finite-dimensional Lie algebras and the category of finite-dimensional modules, the whole subject, both in results and methods employed, becames quite different (one could mention the classical Whitehead Lemmata for semisimple Lie algebras in characteristic zero and Dzhumadildaev–Farnsteiner– Strade non-vanishing result in characteristic p). We will not touch this subject here. Obviously, a free Lie algebra, due to its universal property, has cohomological dimension 1. The question is whether the converse is true. This was asked several times, among them by Bourbaki [B, footnote to Ex. II.2.9] and in [MZ, Problem 28.11]. In the late 60s, Stallings [St1], [St2] (for the case of finitely-generated groups) and Swan [Sw] (for the general case) proved that a group G of cohomological dimension 1 is free (where cohomology understood in the category of ZG-modules). Their initial reasonings were based on the notion of ends of topological spaces and contained a good deal of topology. A streamlined and purely algebraic proof may be found in [Co], a more recent nice surveys of these and related results and ideas may be found in [Ca, §6] and [E], and a guide through the original Stallings–Swan proof may be found in [CZ, §6.2] and [St3, §4-5] (the latter one, due to Stallings himself, also contains a very clear introduction to the theory of ends). Ends in different topological and algebraic settings are discussed also in [HR] (without any connection to Stallings–Swan theorem), and probably the study of the latter book may reveal new approaches. Dunwoody [D] extended the Stallings-Swan theorem to the category of RG-modules for arbitary coefficients ring R.