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.

# more noncommutative manifolds

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