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.


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


Zbl 0634.20015
Full Text: DOI


[1] B. Bowditch, Notes on Gromov’s hyperbolicity criterion for path-metric spaces , Group theory from a geometrical viewpoint (Trieste, 1990) eds. Ghys, Haefliger and Verjosvski, World Sci. Publishing, River Edge, NJ, 1991, pp. 64-167. · Zbl 0843.20031
[2] J. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups , Geom. Dedicata 16 (1984), no. 2, 123-148. · Zbl 0606.57003
[3] C. Champetier, Petite simplification dans les groupes hyperboliques , Ann. Fac. Sci. Toulouse Math. (6) 3 (1994), no. 2, 161-221, Propriétés statistiques des groupes de présentation finie, à paraître dans Adv. Math. · Zbl 0803.53026
[4] M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrie et théorie des groupes , Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990. · Zbl 0727.20018
[5] S. Gersten and H. Short, Small cancellation theory and automatic groups , Invent. Math. 102 (1990), no. 2, 305-334. · Zbl 0714.20016
[6] E. Ghys and P. de la Harpe, Sur les groupes hyperboliques d’après Mikhael Gromov , Progress in Mathematics, vol. 83, Birkhäuser Boston Inc., Boston, MA, 1990. · Zbl 0731.20025
[7] M. Gromov, Hyperbolic groups , Essays in Group Theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer-Verlag, New York, 1987, pp. 75-263. · Zbl 0634.20015
[8] P. de la Harpe and A. Valette, La propriété \((T)\) de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger) , Astérisque (1989), no. 175, 158. · Zbl 0759.22001
[9] R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory , Springer-Verlag, Berlin, 1977. · Zbl 0368.20023
[10] 1 A. Yu. Olshanskiĭ, On residualing homomorphisms and \(G\)-subgroups of hyperbolic groups , Internat. J. Algebra Comput. 3 (1993), no. 4, 365-409. · Zbl 0830.20053
[11] 2 A. Yu. Olshanskiĭ, Periodic quotient groups of hyperbolic groups , Mat. Sb. 182 (1991), no. 4, 543-567.
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