Isoperimetric functions of groups and computational complexity of the word problem

@article{JCBirget1998IsoperimetricFO,
  title={Isoperimetric functions of groups and computational complexity of the word problem},
  author={J.-C.Birget and A. Olshanskii and E. Rips and M.Sapir},
  journal={Annals of Mathematics},
  year={1998},
  volume={156},
  pages={467-518}
}
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]. 
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