# Cohomological dimension of finitely presented group

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?

• A concrete example: Take figure 8 knot (or any nontrivial 2-bridge knot). Then the fundamental group of the complement is 2-generated and has cohomological dimension 2. – Misha Jun 14 '12 at 16:22
• Or any Baumslag--Solitar group (presentation $\langle a,b\mid ba^mb^{-1}=b^n\rangle$ ). – HJRW Jun 14 '12 at 18:02
• Oh, I see that Lee Mosher wrote something similar in a comment below. The bottom line is that there is a huge bestiary of groups of cohomological dimension two. – HJRW Jun 14 '12 at 18:06

• Ok, I guess it follows from the Magnus-Moldovanskii hierarchy. Wise's recent results also imply the virtual cohomological dimension is $\leq 2$ in the presence of torsion. – Ian Agol Jun 14 '12 at 22:53
• Here's a concrete example of this type, one of my favorite groups: the solvable Baumslag-Solitar group $BS(1,n) = <a,t | tat^{-1}=a^n>$. – Lee Mosher Jun 14 '12 at 16:31