Groupes aléatoires [d'après Misha Gromov, ...]

Ghys, Étienne

Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Exposé no. 916, pp. 173-204.

Quelles sont les propriétés d’un groupe de présentation finie “tiré au hasard” ? La réponse à cette question dépend bien entendu de la méthode choisie pour le tirage au sort. On peut par exemple fixer $n$ générateurs et choisir $p$ relations aléatoirement parmi les mots de longueur $L$, puis faire tendre $L$ vers l’infini. On peut aussi choisir un graphe fini, étiqueter aléatoirement ses arêtes par des générateurs, et considérer le groupe engendré par ces générateurs, soumis aux relations lues sur les cycles du graphe. Dans cet exposé, je voudrais présenter des travaux de M. Gromov qui permettent de répondre à ces questions et qui mettent en évidence l’existence de groupes de présentation finie aux propriétés étonnantes.

What are the properties of a finitely presented group “chosen at random”? The answer to this question depends on the method of sorting a group at random. One could fix the number $n$ of generators and choose $p$ relators at random among words of length $L$, and then let $L$ go to infinity. One could also choose some finite graph, label its edges randomly by generators, and consider the group generated by these generators subject to the relations read on the cycles of the graph. In this talk, I would like to introduce the reader to some works of M. Gromov answering this kind of questions. These methods produce examples of finitely presented groups with surprising properties.

[Al91] J. M. Alonso et al. - “Notes on word hyperbolic groups”, in Group theory from a geometrical viewpoint (Trieste 1990), World Sci. Publishing, River Edge, NJ, 1991, p. 3-63. | MR | Zbl

[Ar97] G. Arzhantseva - “Sur les groupes dont les sous-groupes ayant un nombre fixé de générateurs sont libres”, Fundam. Prikl. Mat. 3 (1997), no. 3, p. 675-683, (en russe). | Zbl

[Ar98] -, “Generic properties of finetely presented groups and Howson's theorem”, Comm. Algebra 26 (1998), no. 11, p. 3783-3792. | MR | Zbl

[AO96] G. Arzhantseva & A. Yu. Ol'Shanskiĭ - “Généricité de la classe des groupes dont les sous-groupes ayant moins de générateurs sont libres”, Mat. Zametki 59 (1996), no. 4, p. 489-496, (en russe). | Zbl

[BS97] W. Ballmann & J. Swiatkowski - “On $L^2$-cohomology and property (T) for automorphism groups of polyhedral cell complexes”, Geom. Funct. Anal. 7 (1997), no. 4, p. 615-645. | MR | Zbl

[Bo01] B. Bollobás - Random graphs, 2e 'ed., Cambridge Studies in Advanced Mathematics, vol. 73, Cambridge University Press, Cambridge, 2001. | MR | Zbl

[Bo75] A. Borel - “Cohomologie de certains groupes discrets et laplacien $p$-adique (d’après H. Garland)”, in Séminaire Bourbaki (1973/1974), Lect. Notes in Math., vol. 431, Springer, Berlin, 1975, exp. no 437, p. 12-35. | EuDML | Numdam | MR | Zbl

[BH99] M. R. Bridson & A. Haefliger - Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, vol. 319, Springer-Verlag, Berlin, 1999. | MR | Zbl

[Ch91] C. Champetier - “Propriétés génériques des groupes de présentation finie”, Thèse de doctorat, Université de Lyon I, décembre 1991.

[Ch93] -, “Cocroissance des groupes à petite simplification”, Bull. London Math. Soc. 25 (1993), no. 5, p. 438-444. | MR | Zbl

[Ch94] -, “Petite simplification dans les groupes hyperboliques”, Ann. Fac. Sci. Toulouse Math. (6) 3 (1994), no. 2, p. 161-221. | EuDML | Numdam | MR | Zbl

[Ch95] -, “Propriétés statistiques des groupes de présentation finie”, Adv. Math. 116 (1995), no. 2, p. 197-262. | MR | Zbl

[Ch00] -, “L'espace des groupes de type fini”, Topology 39 (2000), no. 4, p. 657-680. | MR | Zbl

[CM82] B. Chandler & W. Magnus - The history of combinatorial group theory, Studies in the History of Mathematics and Physical Sciences, vol. 9, Springer-Verlag, New York, 1982, A case study in the history of ideas. | MR | Zbl

[CDP90] M. Coornaert, T. Delzant & A. Papadopoulos - Géométrie et théorie des groupes, Lect. Notes in Math., vol. 1441, Springer-Verlag, Berlin, 1990, Les groupes hyperboliques de Gromov. | MR | Zbl

[De96] T. Delzant - “Sous-groupes distingués et quotients des groupes hyperboliques”, Duke Math. J. 83 (1996), no. 3, p. 661-682. | MR | Zbl

[De03] -, “Mesoscopic curvature and very small cancellation theory (after M. Gromov)”, manuscrit, 2003.

[Ga73] H. Garland - “$p$-adic curvature and the cohomology of discrete subgroups of $p$-adic groups”, Ann. of Math. (2) 97 (1973), p. 375-423. | MR | Zbl

[Gh90] E. Ghys - “Les groupes hyperboliques”, in Séminaire Bourbaki (1989/1990), Astérisque, vol. 189-190, Société Mathématique de France, Paris, 1990, exp. no 722, p. 203-238. | EuDML | Numdam | MR | Zbl

[GhH90] E. Ghys & P. De La Harpe - Sur les groupes hyperboliques, d'après M. Gromov, Progress in Math., vol. 83, Birkhäuser, Boston, 1990. | MR | Zbl

[Gri85] R. Grigorchuk - “Degrees of growth of finitely generated groups and the theory of invariant means”, Mathematics of the USSR Izvestiya 25 (1985), no. 2, p. 259-300. | MR | Zbl

[Gr81] M. Gromov - “Hyperbolic manifolds, groups and actions”, in Riemann surfaces and related topics : Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y. 1978), Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, p. 183-213. | MR | Zbl

[Gr84] -, “Infinite groups as geometric objects”, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Varsovie 1983), PWN, Varsovie, 1984, p. 385-392. | MR

[Gr87] -, “Hyperbolic groups”, in Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, p. 75-263. | MR | Zbl

[Gr93] -, “Asymptotic invariants of infinite groups”, in Geometric group theory, Vol. 2 (Sussex 1991), London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, 1993, p. 1-295. | MR | Zbl

[Gr01a] -, “CAT$\left(\kappa \right)$-spaces : construction and concentration”, Geom. i Topol. 7 (2001), p. 100-140, 299-300, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) vol. 280. | MR

[Gr01b] -, “Mesoscopic curvature and hyperbolicity”, in Global differential geometry : the mathematical legacy of Alfred Gray (Bilbao 2000), Contemp. Math., vol. 288, Amer. Math. Soc., Providence, RI, 2001, p. 58-69. | MR | Zbl

[Gr01c] -, “Small cancellation, unfolded hyperbolicity, and transversal measures”, in Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math., vol. 38, Enseignement Math., Genève, 2001, p. 371-399. | MR | Zbl

[Gr03] -, “Random walk in random groups”, Geom. Funct. Anal. 13 (2003), no. 1, p. 73-146. | MR | Zbl

[Ha00] P. De La Harpe - Topics in geometric group theory, Chicago Lectures in Mathematics Series, 2000. | MR | Zbl

[HV89] P. De La Harpe & A. Valette - La propriété $\left(T\right)$ de Kazhdan pour les groupes localement compacts, Astérisque, no. 175, Société Mathématique de France, Paris, 1989, appendice de Marc Burger. | Numdam | Zbl

[HLS02] N. Higson, V. Lafforgue & G. Skandalis - “Counterexamples to the Baum-Connes conjecture”, Geom. Funct. Anal. 12 (2002), p. 330-354. | MR | Zbl

[HR00] N. Higson & J. Roe - “Amenable group actions and the Novikov conjecture”, J. reine Angew. Math. 519 (2000), p. 143-153. | MR | Zbl

[IO96] S. Ivanov & A. Yu. Ol'Shanskiĭ - “Hyperbolic groups and their quotients of bounded exponents”, Trans. Amer. Math. Soc. 348 (1996), no. 6, p. 2091-2138. | MR | Zbl

[KS02] I. Kapovich & P. Schupp - “Genericity, the Arzhantseva-Ol'shanskii method and the isomorphism problem for one-relator groups”, Prépublication, ArXiv:math.GR/0210307, octobre 2002. | Zbl

[Ke59] H. Kesten - “Symmetric random walks on groups”, Trans. Amer. Math. Soc. 92 (1959), p. 336-354. | MR | Zbl

[Lu94] A. Lubotzky - Discrete groups, expanding graphs and invariant measures, Progress in Math., vol. 125, Birkhäuser Verlag, Basel, 1994, appendice de Jonathan D. Rogawski. | MR | Zbl

[LS01] R. C. Lyndon & P. E. Schupp - Combinatorial group theory, Classics in Mathematics, Springer-Verlag, 2001, nouveau tirage de l'édition de 1977. | MR | Zbl

[Ne37] B. H. Neumann - “Some remarks on infinite groups”, J. London Mat. Soc. 12 (1937), p. 120-127. | JFM

[Oll03a] Y. Ollivier - “Sharp phase transition theorems for hyperbolicity of random groups”, Geom. Funct. Anal. (2003), à paraître, prépublication, ArXiv:math.GR/0301187. | MR | Zbl

[Oll03b] -, “On a small cancellation theorem of Gromov”, manuscrit, janvier 2003.

[Oll03c] -, “Critical densities for random quotients of hyperbolic groups”, C. R. Acad. Sci. Paris Sér. I Math. 336 (2003), no. 5, p. 391-394. | MR | Zbl

[Ols92a] A. Yu. Ol'Shanskiĭ - “Almost every group is hyperbolic”, Internat. J. Algebra Comput. 2 (1992), no. 1, p. 1-17. | MR | Zbl

[Ols92b] -, “Periodic factor groups of hyperbolic groups”, Mathematics of the USSR Sbornik 72 (1992), p. 519-541. | MR

[Pa98] P. Pansu - “Formules de Matsushima, de Garland et propriété (T) pour des groupes agissant sur des espaces symétriques ou des immeubles”, Bull. Soc. Math. France 126 (1998), no. 1, p. 107-139. | EuDML | Numdam | MR | Zbl

[Pa96] P. Papasoglu - “An algorithm detecting hyperbolicity”, in Geometric and computational perspectives on infinite groups (Minneapolis, MN and New Brunswick, NJ 1994), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 25, Amer. Math. Soc., Providence, RI, 1996, p. 193-200. | MR | Zbl

[Se92] Z. Sela - “Uniform embeddings of hyperbolic groups in Hilbert spaces”, Israel J. Math. 80 (1992), no. 1-2, p. 171-181. | MR | Zbl

[Si03] L. Silberman - “Addendum to : “Random walk in random groups” [Geom. Funct. Anal. 13 (2003), no. 1, 73-146 ; MR1978492] by M. Gromov”, Geom. Funct. Anal. 13 (2003), no. 1, p. 147-177. | MR | Zbl

[Sk00] G. Skandalis - “Progrès récents sur la conjecture de Baum-Connes. Contribution de Vincent Lafforgue”, in Séminaire Bourbaki (1999/2000), Astérisque, vol. 276, Société Mathématique de France, Paris, 2002, exp. no 829, p. 105-135. | EuDML | Numdam | MR | Zbl

[SZ94] G. Stuck & R. J. Zimmer - “Stabilizers for ergodic actions of higher rank semisimple groups”, Annals Math. 139 (1994), p. 723-747. | MR | Zbl

[TV99] S. Thomas & B. Velickovic - “On the complexity of the isomorphism relation for finitely generated groups”, J. Algebra 217 (1999), no. 1, p. 352-373. | MR | Zbl

[Va02] A. Valette - “Nouvelles approches de la propriété (T) de Kazhdan”, in Séminaire Bourbaki (2002/2003), exp. no 913, ce volume. | EuDML | Numdam | Zbl

[Yu00] G. Yu - “The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space”, Invent. Math. 139 (2000), no. 1, p. 201-240. | MR | Zbl

[Zuk96] A. Żuk - “La propriété (T) de Kazhdan pour les groupes agissant sur les polyèdres”, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), no. 5, p. 453-458. | MR | Zbl

[Zuk03] -, “Property (T) and Kazhdan constants for discrete groups”, Geom. Funct. Anal. 13 (2003), no. 3, p. 643-670. | MR | Zbl