Actions of finitely generated groups on $ℝ$-trees

Guirardel, Vincent

Annales de l'Institut Fourier, Tome 58 (2008) no. 1, pp. 159-211.

On étudie les actions de groupes de type fini sur des arbres réels sous certaines hypothèses de stabilité. On démontre que soit le groupe se scinde au dessus de sous-groupes contrôlés (fixant un arc en particulier), soit que l’action peut être obtenue par recollement d’actions simples : actions sur des arbres simpliciaux, actions sur des droites, et actions venant de feuilletages mesurés sur des $2$-orbifolds. Ceci étend des résultats de Sela et de Rips-Sela. Cependant, leurs résultats sont mal énoncés, et on donne un contrexemple à leurs énoncés.

La preuve repose sur une version étendue du Lemme de Scott qui est intéressante en soi. Cet énoncé affirme que si un groupe $G$ est une limite directe de groupes ayant des scindements compatibles en un sens convenable, alors $G$ se scinde.

We study actions of finitely generated groups on $ℝ$-trees under some stability hypotheses. We prove that either the group splits over some controlled subgroup (fixing an arc in particular), or the action can be obtained by gluing together actions of simple types: actions on simplicial trees, actions on lines, and actions coming from measured foliations on $2$-orbifolds. This extends results by Sela and Rips-Sela. However, their results are misstated, and we give a counterexample to their statements.

The proof relies on an extended version of Scott’s Lemma of independent interest. This statement claims that if a group $G$ is a direct limit of groups having suitably compatible splittings, then $G$ splits.

Classification : 20E08, 20F65, 20E06
Keywords: R-tree, splitting of group, Rips theory
Mot clés : arbre réel, décomposition de groupe, théorie de Rips

[1] Alibegovic, Emina Makanin-Razborov diagrams for limit groups (2004) (math.GR/0410198)

[2] Belegradek, Igor; Szczepanski, Andrzej Endomorphisms of relatively hyperbolic groups (2005) (math.GR/0501321)

[3] Bestvina, M.; Feighn, M. Outer Limits (Preprint)

[4] Bestvina, M.; Feighn, M. Bounding the complexity of simplicial group actions on trees, Invent. Math., Volume 103 (1991) no. 3, pp. 449-469 | DOI | MR | Zbl

[5] Bestvina, M.; Feighn, M. A combination theorem for negatively curved groups, J. Differential Geom., Volume 35 (1992) no. 1, pp. 85-101 | MR | Zbl

[6] Bestvina, M.; Feighn, M. Stable actions of groups on real trees, Invent. Math., Volume 121 (1995) no. 2, pp. 287-321 | DOI | MR | Zbl

[7] Chiswell, Ian Introduction to $\Lambda$-trees, World Scientific Publishing Co. Inc., River Edge, NJ, 2001 | MR | Zbl

[8] Cohen, M. M.; Lustig, M. Very small group actions on $ℝ$-trees and Dehn twist automorphisms, Topology, Volume 34 (1995) no. 3, pp. 575-617 | DOI | MR | Zbl

[9] Cooperative, Group Theory MAGNUS, Computational package for exploring infinite groups, version 4.1.3 beta, 2005 (G.Baumslag director)

[10] Culler, M.; Morgan, J. W. Group actions on $ℝ$-trees, Proc. London Math. Soc. (3), Volume 55 (1987) no. 3, pp. 571-604 | DOI | MR | Zbl

[11] Delzant, Thomas Sur l’accessibilité acylindrique des groupes de présentation finie, Ann. Inst. Fourier (Grenoble), Volume 49 (1999) no. 4, pp. 1215-1224 | DOI | Numdam | Zbl

[12] Druţu, Cornelia; Sapir, Mark Groups acting on tree-graded spaces and splittings of relatively hyperbolic group (2006) (math.GR/0601305)

[13] Dunwoody, M. J. Folding sequences, The Epstein birthday schrift, Geom. Topol., Coventry, 1998, p. 139-158 (electronic) | MR | Zbl

[14] Gaboriau, D.; Levitt, G.; Paulin, F. Pseudogroups of isometries of $ℝ$ and Rips’ theorem on free actions on $ℝ$-trees, Israel J. Math., Volume 87 (1994) no. 1-3, pp. 403-428 | DOI | Zbl

[15] Gaboriau, D.; Levitt, G.; Paulin, F. Pseudogroups of isometries of $ℝ$: reconstruction of free actions on $ℝ$-trees, Ergodic Theory Dynam. Systems, Volume 15 (1995) no. 4, pp. 633-652 | DOI | MR | Zbl

[16] Groves, Daniel Limit groups for relatively hyperbolic groups. II. Makanin-Razborov diagrams, Geom. Topol., Volume 9 (2005), p. 2319-2358 (electronic) | DOI | MR | Zbl

[17] Guirardel, V. Actions de groupes sur des arbres réels et dynamique dans la frontière de l’outre-espace, Université Toulouse III, jan (1998) (Ph. D. Thesis)

[18] Guirardel, V. Approximations of stable actions on $ℝ$-trees, Comment. Math. Helv., Volume 73 (1998) no. 1, pp. 89-121 | DOI | MR | Zbl

[19] Guirardel, V. Limit groups and groups acting freely on ${ℝ}^n$-trees, Geom. Topol., Volume 8 (2004), p. 1427-1470 (electronic) | DOI | MR | Zbl

[20] Guirardel, V. Cœur et nombre d’intersection pour les actions de groupes sur les arbres, Ann. Sci. École Norm. Sup. (4), Volume 38 (2005) no. 6, pp. 847-888 | Numdam | Zbl

[21] Gusmão, Paulo Feuilletages mesurés et pseudogroupes d’isométries du cercle, J. Math. Sci. Univ. Tokyo, Volume 7 (2000) no. 3, pp. 487-508 | Zbl

[22] Imanishi, H. On codimension one foliations defined by closed one-forms with singularities, J. Math. Kyoto Univ., Volume 19 (1979) no. 2, pp. 285-291 | MR | Zbl

[23] Kapovich, I.; Weidmann, R. Acylindrical accessibility for groups acting on $ℝ$-trees, Math. Z., Volume 249 (2005) no. 4, pp. 773-782 | DOI | MR | Zbl

[24] Levitt, Gilbert La dynamique des pseudogroupes de rotations, Invent. Math., Volume 113 (1993) no. 3, pp. 633-670 | DOI | MR | Zbl

[25] Levitt, Gilbert Graphs of actions on $ℝ$-trees, Comment. Math. Helv., Volume 69 (1994) no. 1, pp. 28-38 | DOI | MR | Zbl

[26] Levitt, Gilbert; Paulin, Frédéric Geometric group actions on trees, Amer. J. Math., Volume 119 (1997) no. 1, pp. 83-102 | DOI | MR | Zbl

[27] Morgan, John W. Ergodic theory and free actions of groups on $ℝ$-trees, Invent. Math., Volume 94 (1988) no. 3, pp. 605-622 | DOI | MR | Zbl

[28] Morgan, John W.; Shalen, Peter B. Valuations, trees, and degenerations of hyperbolic structures. I, Ann. of Math. (2), Volume 120 (1984) no. 3, pp. 401-476 | DOI | MR | Zbl

[29] Paulin, Frédéric Topologie de Gromov équivariante, structures hyperboliques et arbres réels, Invent. Math., Volume 94 (1988) no. 1, pp. 53-80 | DOI | MR | Zbl

[30] Rips, E.; Sela, Z. Structure and rigidity in hyperbolic groups. I, Geom. Funct. Anal., Volume 4 (1994) no. 3, pp. 337-371 | DOI | MR | Zbl

[31] Scott, G. P. Finitely generated $3$-manifold groups are finitely presented, J. London Math. Soc. (2), Volume 6 (1973), pp. 437-440 | DOI | MR | Zbl

[32] Sela, Z. Acylindrical accessibility for groups, Invent. Math., Volume 129 (1997) no. 3, pp. 527-565 | DOI | MR | Zbl

[33] Sela, Z. Endomorphisms of hyperbolic groups. I. The Hopf property, Topology, Volume 38 (1999) no. 2, pp. 301-321 | DOI | MR | Zbl

[34] Sela, Z. Diophantine geometry over groups. I. Makanin-Razborov diagrams, Publ. Math. Inst. Hautes Études Sci., Volume 93 (2001), pp. 31-105 | Numdam | MR | Zbl

[35] Sela, Z. Diophantine geometry over groups VII: The elementary theory of a hyperbolic group (2002) (http://www.ma.huji.ac.il/~zlil)

[36] Sela, Z. Diophantine geometry over groups. VI. The elementary theory of a free group, Geom. Funct. Anal., Volume 16 (2006) no. 3, pp. 707-730 | MR | Zbl

[37] Serre, J-P. Arbres, amalgames, ${\mathrm{S}\mathrm{L}}_2$, Société Mathématique de France, Paris, 1977 (Rédigé avec la collaboration de Hyman Bass, Astérisque, No. 46) | Numdam | Zbl

[38] Shalen, Peter B. Dendrology and its applications, Group theory from a geometrical viewpoint (Trieste, 1990), World Sci. Publishing, River Edge, NJ, 1991, pp. 543-616 | MR | Zbl

[39] Skora, Richard Combination theorems for actions on trees (1989) (preprint)

[40] Stallings, John R. Topology of finite graphs, Invent. Math., Volume 71 (1983) no. 3, pp. 551-565 | DOI | MR | Zbl

[41] Swarup, Gadde Delzant’s variation on Scott complexity (2004) (arXiv:math.GR/0401308)