Groups with finite 2-dimensional Eilenberg–Mac Lane spaces

This theorem seems to come tantalizingly close to supplying counter-examples to the Eilenberg–Ganea conjecture, which asserts that any group of cohomological dimension 2 admits a (not necessarily finite) 2-dimensional Eilenberg–Mac Lane space. Indeed, I don’t know of any example of a finitely-presented group with an infinite 2-dimensional K(Γ, 1) but not a finite one, but such things surely exist. I must also confess that I don’t know of an inefficient group of cohomological dimension 2, though Lustig has constructed torsion-free inefficient groups [9]. Hillman’s proof depends on some heavy-duty technical machinery. It is a corollary of Eckmann’s solution of the Bass conjecture for groups of cohomological dimension 2, which itself rests on Stallings’s ends theorem and some serious cyclic homology. One might hope that there would be a more conceptual proof, but since we are forced to leave the congenial world of homology (after all, the theorem is asserting that homology isn’t a sufficiently subtle invariant to detect some topological property) we should not be surprised to have to scrabble around and grab whatever tools we can muster. In these notes I focus on the input from cyclic homology, largely because [8] is an exhilarating and inspiring paper, and it duly left me exhilarated, and indeed inspired. The Bass conjecture also seems to have become quite fashionable again nowadays, amidst the general frenzy around the Baum–Connes conjecture. § 1 develops the basics of cyclic homology, and proves Burghelea’s theorem [2] computing the cyclic homology of group algebras kΓ for commutative fields k containing Q. The presentation of this largely follows [11] (and if Weibel is easier to read than something. . . ). § 2 gives Eckmann’s proof of the Bass conjecture for groups of cohomological dimension 2, assuming the necessary results from group cohomology. (Eckmann’s paper contains stronger results than those given here—I just give what’s needed for Hillman’s result.) § 3 is very short, and gives Hillman’s proof. Please be aware that these notes are for my benefit, and so are probably rather uneven, labouring those things that are new to me and skipping over things I’m more comfortable with.