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 non-deterministic 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 $H$. 
J.-C.Birget, A.Yu. Olshanskii, E. Rips, M.Sapir