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Distinguished subgroups and quotients of hyperbolic groups. (Sous-groupes distingués et quotients des groupes hyperboliques.) (French) Zbl 0852.20032
The author studies normal subgroups of hyperbolic groups and quotients of hyperbolic groups. He develops ideas of classical small-cancellation theory and generalizes them to the setting of hyperbolic groups. In particular, he proves a lemma which is analogous to Greendlinger’s lemma (which is a fundamental lemma in classical small cancellation theory).
To state the main results of the paper, we let $$G$$ be a hyperbolic group equipped with a word-metric. For every element $$f$$ in $$G$$, $$|f|$$ is the minimal length of the elements in $$G$$ which are conjugate to $$f$$. The stable norm $$|f|$$ of $$f$$ is the limit as $$n$$ tends to infinity of the quantity $${1\over n}|f^n|$$. Thus, the stable norm is a conjugacy invariant.
The main results are the following: Theorem I. Let $$G$$ be a non-elementary $$\delta$$-hyperbolic group. Then there exists an integer $$N$$ such that for every family $$f_1,\dots,f_n$$ of elements of $$G$$ satisfying $$|f_i|=|f_j|\geq 1000\delta$$ for all $$i$$ and $$j$$, the normal subgroup $$\langle f^N_i\rangle$$ generated by the $$N$$-th powers of these elements is a free group. Furthermore, for every integer $$k$$, the quotient group $$G/\langle f^{kN}_i\rangle$$ is hyperbolic. Theorem II. Let $$\mathcal R$$ be a collection of elements in $$G$$ satisfying the small cancellation condition $$C'(l)$$. Then the normal subgroup $$\langle{\mathcal R}\rangle$$ generated by $$\mathcal R$$ is free and the quotient group $$G/\langle{\mathcal R}\rangle$$ is hyperbolic. (The small cancellation condition $$C'(l)$$ is one of the main technical conditions that are defined in the paper).
The author, while using many ideas of M. Gromov contained in his paper Hyperbolic groups [Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)] gives a precise meaning to several statements and arguments of Gromov which were obscure. In particular, in an appendix to the paper, the author gives a counter-example to Theorem 5.3.E of Gromov’s paper and answers negatively (by the same counter-example) question 5.3.D of that paper.

##### MSC:
 20F65 Geometric group theory 20F06 Cancellation theory of groups; application of van Kampen diagrams 20F05 Generators, relations, and presentations of groups 20E07 Subgroup theorems; subgroup growth
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##### References:
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