Just recently I've stumbled across Warren Dicks' book *Groups, trees and projective modules* (1980) and I was pretty stunned. I know nothing of group cohomology, but I gather the "tree" component is a special case of space studied in general group cohomology.

The results that caught my attention were:

The augmentation ideal of $R[G]$ is projective iff $G$ has cohomological dimension at most $1$, and it has cohomological dimension exactly $1$ iff it is an infinite group of cohomological dimension $1$.

The augmentation ideal of $R[G]$ is projective iff $G$ is the fundamental group of a graph of finite groups having order invertible in $R$.

For me this resonated with two other results I know well about group rings:

(Renault) $R[G]$ is right self-injective iff $R$ is right self-injective and $G$ is finite.

and

(Connell) $R[G]$ is von Neumann regular iff 1) $R$ is von Neumann regular; 2) $G$ is locally finite; and 3) the order of every finite subgroup of $G$ is invertible in $R$.

Suppose from here on that $R[G]$ is von Neumann regular, $G$ is (at most) countable, and you don't know Connell's proof about regular group rings.

Then the augmentation ideal is countably generated, and by a result of Kaplansky the augmentation ideal must be projective. Apparently now $G$ is "the fundamental group of a graph of finite groups having order invertible in $R$."

**First question:** is it obvious somehow that $G$ is locally finite and that the groups which are vertices of the graph of groups tell us about the finite subgroups of $G$ and the invertibility of their orders? (I.e., can you recover Connell's theorem from Dunwoody's theorem?)

**Second question:** now additionally assume $R[G]$ is right self-injective and you don't know Renault's result. Can right self-injectivity be interpreted in this context to explain why $G$ is finite, i.e. $G$ has cohomological dimension $0$? (I.e., can you recover Renault's theorem in the special case of VNR right self-injective rings from Dunwoody's theorem by explaining why injectivity of $R[G]$ decides that $G$ is finite?)

I'm just probing around here seeing if I can get some connection between the two disciplines.