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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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