$\langle a,b \,|\, ab = ba\rangle$ is the fundamental group of the torus. Consider $\langle x,y,z\,|\, xyz = zyx\rangle$. We have the homeomorphism $f(a) = zx, f(b) = zy^{-1}$. Is there an isomorphism? I think there should be because $\langle x,y,z\,|\, xyz = zyx\rangle$ is the fundamental group of the object from this question which is supposed to be homeomorphic to the torus.
Am I misunderstanding something? If not how can I find the explicit isomorphism, in particular I want to know what $x,y,z$ are mapped to.
Edit: @Desperado comment resolved my confusion, the hexagon I was thinking about had all vertices identified to each other while the one in the post I linked does not. Thanks!
They are definitely not isomorphic, as they have different abelianization. The group ⟨x,y,z|xyz=zyx⟩ is not the fundamentl group of the "hexagon" in the linked question: the vertices are not all the same point, but 3 are P and 3 are Q≠P."
The group $G=\langle x,y,z\mid xyz=zyx\rangle$ is non-abelian, hence it cannot be isomorphic to the group $\langle a,b\mid ab=ba\rangle \cong \Bbb Z\times \Bbb Z$.