Good evening
I was trying to prove that the commutator [F2,F2] of the free group F2 is not finitely generated by using covering spaces (i have to admit that this is the idea of a friend) it seems that F2 is the fundamental group of the plane with two points removed.
I al thinking of linking [F2,F2] to the kernel of the homomorphism that we use to prove that the quotient π1(ZxZ)/[π1(ZxZ),π1(ZxZ)] isomorphic to H1(ZxZ).
The homology group here is the free abelian group with two generators: we can visualise it as the vertices of ZxZ. we can visualise the map from F2 (with generators a and b) to ZxZ as follows: to go from the origin of the grid to a point in ZxZ we move horizontally and vertically by units to go from O to (1,2) we go upward 1 right 1 then upward 1 we model that movement by "bab" we can move right and then upward twice which makes it "abb" then we notice that we have a homomorphism from Fé to ZxZ with a kernel which is exactly [F2,F2].
Any ideas?