This is really a comment rather than an answer, but perhaps the answer to your question follows from doing a mathscinet search? I did one and I found the following reference:
Dunwoody, M. J.
Accessibility and groups of cohomological dimension one.
Proc. London Math. Soc. (3) 38 (1979), no. 2, 193–215.
In this important and exciting paper the author obtains the structure of groups of cohomological dimension one over an arbitrary (commutative) ring with unity. Theorem 1.1: cdRG≤1 if and only if G is isomorphic to the fundamental group of a graph of groups in which every vertex group is finite with no R-torsion.(A finite group has no R-torsion if its order is invertible in R.) This extends the results of J. R. Stallings [Ann. of Math. (2) 88 (1968), 312--334; MR0228573 (37 #4153)] and R. G. Swan [J. Algebra 12 (1969), 585--610; MR0240177 (39 #1531)] that if cdRG≤1 and G is torsion-free then G is free, and also the results of various authors on free-by-finite groups.
The methods are an ingenious combination of Bass-Serre theory with a relative version of the theory of accessible groups due to C. Bamford and the author [J. Pure Appl. Algebra 7 (1976), no. 3, 333--346; MR0399271 (53 #3122)], and the reviewer's approach to the Stallings-Swan theory by means of almost invariant subsets [Groups of cohomological dimension one, Lecture Notes in Math., Vol. 245, Springer, Berlin, 1972; MR0344359 (49 #9098)].
In the course of the paper the author gives what is likely to be the best possible proof of Stallings' structure theorem for groups with more than one end [Stallings, op. cit.]. It has been known for some time that this theorem can be expressed by saying that the group acts on a tree with suitable properties. The reviewer spent much time trying to find the relevant tree; the set of edges of the tree was fairly obvious but he had no success in finding the vertex set. The author's solution is brilliantly simple. There is no need to determine the vertex set! His tree theorem (Theorem 2.1) gives necessary and sufficient conditions for a set to be the set of edges of a tree. These conditions, for a group with more than one end, are little more than previously known results about almost invariant sets.
The following might also have something to do with your question:
Cohen, Daniel E.
Groups of cohomological dimension one.
Lecture Notes in Mathematics, Vol. 245. Springer-Verlag, Berlin-New York, 1972. v+99 pp
The object of these well-written notes is to give a completely self-contained account of the following theorems. Theorem A: A torsion-free group of cohomological dimension one over some ring with unit is free. Theorem B: A torsion-free group containing a free subgroup of finite index is free. Theorem C: Let H be a subgroup of finite index in a torsion-free group G; then G and H have the same cohomological dimension. Theorem D: Let H be a subgroup of a free group G; then H is a free factor of G if and only if IHG is a direct summand in IG, the augmentation ideal of G.
Theorems A and B were proved by J. R. Stallings [Ann. of Math. (2) 88 (1968), 312--334; MR0228573 (37 #4153)] for finitely generated groups and by R. G. Swan [J. Algebra 12 (1969), 585--610; MR0240177 (39 #1531)] in the general case. Theorem C is attributed to Serre; Theorem D is a stronger version of a result of Swan [op. cit.].
The presentation of the material differs in many significant details from the papers of Stallings and Swan; among them are the following: (i) the theory of ends, which plays an important rôle in the proof of Theorem A, is given in the algebraic form due to the author [Math. Z. 114 (1970), 9--18; MR0260877 (41 #5497)]; (ii) Stallings's structure theorem for groups with infinitely many ends [Applications of categorical algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968), pp. 124--128, Amer. Math. Soc., Providence, R.I., 1970; MR0255689 (41 #349)] is given a proof combining methods due to M. J. Dunwoody [J. Algebra 12 (1969), 339--344; MR0238931 (39 #291)] and P. C. Oxley [Math. Z. 127 (1972), 265--272; MR0332997 (48 #11322)]; (iii) some of Swan's homological arguments are replaced by more explicit discussions of the augmentation ideal of the group G.
In an appendix proofs of the theorems of Kuroš and Gruško are included.