Groups of homological dimension one

Abstract

Abstract We report on recent work concerning groups of homological dimension one and detail some methods that may be used in order to determine whether these groups are locally free. Introduction Stallings has established in [20] a characterization of finitely generated free groups, as those groups whose cohomological dimension is one. It is very easy to show that a free group has cohomological dimension one. Indeed, if G is a free group then the augmentation ideal I G is a free ZG -module; in fact, if G is freely generated by a subset S, then I G is a free ZG -module on the set ﹛s − 1 : s ∈ S﹜. The essence of Stallings’ theorem is that the converse implication is also true, namely that any finitely generated group of cohomological dimension one is free. Bieri asked in [2] whether a (stronger) homological version of the latter result holds: Is any finitely generated group of homological dimension one free? Shortly after the publication of the proof of Stallings’ theorem, Swan showed that the finite generation hypothesis is redundant therein, by proving that a (not necessarily finitely generated) group G is free if and only if cd G = 1 (cf. [21]). In that direction, we note that Bieri's question may be equivalently formulated as follows: Is any group of homological dimension one locally free? Some interesting results concerning that problem have been obtained in [5] and [11], by embedding the integral group ring ZG of the group G into the associated von Neumann algebra NG and the algebra UG of unbounded operators which are affiliated to NG . We note that a group G is known to be finitely generated if and only if the augmentation ideal I G is a finitely generated ZG -module, whereas G has homological dimension one if and only if I G is a flat ZG -module. In view of the above mentioned result of Stallings and Swan, the freeness of G is equivalent to the projectivity of IG as a ZG -module.