Distinguished subgroups and quotients of hyperbolic groups
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 \textit{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.
scientific article; zbMATH DE number 910487