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. This completes the work started in [26] and [24].

This is the first of two papers devoted to connections between asymptotic functions of groups and computational complexity. One of the main results of this paper states that if for every m the first m digits of a real number α ≥ 4 are computable in time ≤ C2 2 Cm for some constant C > 0 then n α is equivalent ("big O") to the Dehn function of a finitely presented group. The smallest isodiametric function of this group is n 3/4α . On the other hand if n α is equivalent to the Dehn function of a finitely presented group then the first m digits of α are computable in time ≤ C2 2 2 Cm for some constant C. This implies that, say, functions n π+1 , n e 2 and n α for all rational numbers α ≥ 4 are equivalent to the Dehn functions of some finitely presented group and that n π and n α for all rational numbers α ≥ 3 are equivalent to the smallest isodiametric functions of finitely presented groups.Moreover we describe all Dehn functions of finitely presented groups ≻ n 4 as time functions of Turing machines modulo two conjectures:1. Every Dehn function is equivalent to a superadditive function.2. The square root of the time function of a Turing machine is equivalent to the time function of a Turing machine.

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We introduce certain classes of hyperbolic groups according to their possible actions on real trees. Using these classes and results from the theory of (small) group actions on real trees, we study the structure of hyperbolic groups and their automorphism group.-In [Grl] M. Gromov introduced hyperbolic groups and showed how geometric notions, tools, and results, mostly from the theory of negatively curved manifolds, can be adapted to obtain deep and broad algebraic results on the structure of hyperbolic groups and their subgroups. Gromov's paper and a recent work of the second author on the isomorphism problem [Sell stress the need for understanding the structure of automorphisms of hyperbolic groups and more globally the structure of the automorphism group of a hyperbolic group.In this paper we adapt results from the work of the first author on group actions on real trees [R] to the study of hyperbolic groups and their automorphisms. Our approach is an elaboration of the Bestvina-Paulin method ([B], [P]) and we believe that besides the results we obtain our arguments should be applicable for future problems. The results we get serve as key points in our solution to the isomorphism problem [Se2].We start by introducing certain classes of hyperbolic groups in terms of their possible actions on real trees. This classification, although very simple, turns out to be essential in understanding automorphisms and may serve as possible induction steps for future problems. In sections 2 and 3, we bring an immediate application of the Bestvina-Paulin method for the Hopf and co-Hopf properties for certain hyperbolic groups.The automorphism group of a surface group is generated by Dehn twists and inner automorphisms. The notion of a Dehn twist can be generalized The second author was partially supported by an NSF grant.

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