I have a group of cohomological dimension 2 generated by two elements. Is it possible to deduce that the group is commutative or, more generally, does $\mathrm{cd}\ G=2$ imply anything about the relations that $G$ must satisfy?
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The quotient of the free group of rank 2 by a random, long relator has cohomological dimension 2 and is not commutative. 


To complement @Lee's answer, in "Classification of soluble groups of cohomological dimension two", (Math Z, 1979), Dion Gildenhuys does exactly what he claims, so you can see what you get under very strong additional conditions. 

