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?



Cohomological dimension of finitely presented group
The quotient of the free group of rank 2 by a random, long relator has cohomological dimension 2 and is not commutative.


