more noncommutative manifolds

Can it be that one forgets an entire proof because the result doesn’t seem important or relevant at the time? It seems the only logical explanation for what happened last week. Raf Bocklandt asked me whether a classification was known of all group algebras l G which are noncommutative manifolds (that is, which are formally smooth a la Kontsevich-Rosenberg or, equivalently, quasi-free a la Cuntz-Quillen). I said I didn’t know the answer and that it looked like a difficult problem but at the same time it was entirely clear to me how to attack this problem, even which book I needed to have a look at to get started. And, indeed, after a visit to the library borrowing Warren Dicks lecture notes in mathematics 790 “Groups, trees and projective modules” and browsing through it for a few minutes I had the rough outline of the classification. As the proof is basicly a two-liner I might as well sketch it here.
If l G is quasi-free it must be hereditary so the augmentation ideal must be a projective module. But Martin Dunwoody proved that this is equivalent to G being a group acting on a (usually infinite) tree with finite group vertex-stabilizers all of its orders being invertible in the basefield l. Hence, by Bass-Serre theory G is the fundamental group of a graph of finite groups (all orders being units in l) and using this structural result it is then not difficult to show that the group algebra l G does indeed have the lifting property for morphisms modulo nilpotent ideals and hence is quasi-free.
If l has characteristic zero (hence the extra order conditions are void) one can invoke a result of Karrass saying that quasi-freeness of l G is equivalent to G being virtually free (that is, G has a free subgroup of finite index). There are many interesting examples of virtually free groups. One source are the discrete subgroups commensurable with SL(2,Z) (among which all groups appearing in monstrous moonshine), another source comes from the classification of rank two vectorbundles over projective smooth curves over finite fields (see the later chapters of Serre’s Trees). So one can use non-commutative geometry to study the finite dimensional representations of virtually free groups generalizing the approach with Jan Adriaenssens in Non-commutative covers and the modular group (btw. Jan claims that a revision of this paper will be available soon).
In order to avoid that I forget all of this once again, I’ve written over the last couple of days a short note explaining what I know of representations of virtually free groups (or more generally of fundamental algebras of finite graphs of separable l-algebras). I may (or may not) post this note on the arXiv in the coming weeks. But, if you have a reason to be interested in this, send me an email and I’ll send you a sneak preview.

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