Isoperimetric functions of groups and computational complexity of the word problem

We prove that the word problem of a finitely generated group G is in NP (solvable in polynomial time by a nondeterministic Turing machine) if and only if this group is a subgroup of a finitely presented group H with polynomial isoperimetric function. The embedding can be chosen in such a way that G has bounded distortion in H. This completes the work started in [6] and [25].