We introduce a concept of tree-graded metric space and we use it to show quasi-isometry invariance of certain classes of relatively hyperbolic groups, to obtain a characterization of relatively hyperbolic groups in terms of their asymptotic cones, to find geometric properties of Cayley graphs of relatively hyperbolic groups, and to construct the first example of finitely generated group with a continuum of non-π 1 -equivalent asymptotic cones. Note that by a result of Kramer, Shelah, Tent and Thomas, continuum is the maximal possible number of different asymptotic cones of a finitely generated group, provided that the Continuum Hypothesis is true.
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.
We relate the problem of computing the closure of a finitely generated subgroup of the free group in the pro-V topology, where V is a pseudovariety of finite groups, with an extension problem for inverse automata which can be stated as follows: given partial one-to-one maps on a finite set, can they be extended into permutations generating a group in V? The two problems are equivalent when V is extension-closed. Turning to practical computations, we modify Ribes and Zalesskiĭ's algorithm to compute the pro-p closure of a finitely generated subgroup of the free group in polynomial time, and to effectively compute its pro-nilpotent closure. Finally, we apply our results to a problem in finite monoid theory, the membership problem in pseudovarieties of inverse monoids which are Mal'cev products of semilattices and a pseudovariety of groups. Résumé: Nous établissons un lien entre le problème du calcul de l'adhéerence d'un sous-groupe finiment engendré du groupe libre dans la topologie pro-V, oú V est une pseudovariété de groupes finis, et un probléme d'extension pour les automates inversifs qui peut être énoncé de la faç con suivante: étant données des transformations partielles injectives d'un ensemble fini, peuvent-elles être étendues en des permutations qui engendrent un groupe dans V? Les deux problèmes sont équivalents si V est fermée par extensions. Nous intéressant ensuite aux calculs pratiques, nous modifions l'algorithme de Ribes et Zalesskiĭ pour calculer l'adhérence pro-p d'un sous-groupe finiment engendré du groupe libre en temps polynomial et pour calculer effectivement sa clôture pro-nilpotente. Enfin nous appliquons nos résultats à un problème de théorie des monoïdes finis, celui de de l'appartenance dans les pseudovariétés de monoïdes inversifs qui sont des produits de Mal'cev de demi-treillis et d'une pseudovariété de groupes.
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