A combination theorem for cubulation in small cancellation theory over free products  [ Un theorème de combinaison pour les groupes cubulables en théorie de la petite simplification sur des produits libres ]
Annales de l'Institut Fourier, Tome 67 (2017) no. 4, pp. 1613-1670.

Nous montrons qu’un groupe obtenu comme quotient d’un produit libre d’un nombre fini de groupes cubulables en ajoutant un nombre fini de relations satisfaisant la condition de petite simplification C ' (1/6) est lui aussi cubulable. Cela donne une large classe de nouveaux groupes relativement hyperboliques qui peuvent être cubulés, et constitue le premier exemple de théorème de combinaison pour la cubulabilité de groupes relativement hyperboliques ne recquérant aucune hypothèse sur les sous-groupes périphéraux en dehors de leur cubulabilité. Nous obtenons ceci en construisant des structures d’espaces à murs appropriées pour ces groupes, en combinant des murs venant des facteurs libres avec des murs venant du revêtement universel d’un complexe de groupes de dimension 2 associé.

We prove that a group obtained as a quotient of the free product of finitely many cubulable groups by a finite set of relators satisfying the classical C ' (1/6)–small cancellation condition is cubulable. This yields a new large class of relatively hyperbolic groups that can be cubulated, and constitutes the first instance of a cubulability theorem for relatively hyperbolic groups which does not require any geometric assumption on the peripheral subgroups besides their cubulability. We do this by constructing appropriate wallspace structures for such groups, by combining walls of the free factors with walls coming from the universal cover of an associated 2-complex of groups.

Reçu le : 2015-11-27
Révisé le : 2016-04-20
Accepté le : 2016-10-26
Publié le : 2017-09-25
DOI : https://doi.org/10.5802/aif.3118
Classification : 20F06,  20F65,  20F67
Mots clés : actions de groupes sur des complexes cubiques CAT(0), théorie de la petite simplification sur un produit libre, cubulation de groupes.
@article{AIF_2017__67_4_1613_0,
     author = {Martin, Alexandre and Steenbock, Markus},
     title = {A combination theorem for cubulation in small cancellation theory over free products},
     journal = {Annales de l'Institut Fourier},
     pages = {1613--1670},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {4},
     year = {2017},
     doi = {10.5802/aif.3118},
     language = {en},
     url = {archive.numdam.org/item/AIF_2017__67_4_1613_0/}
}
Martin, Alexandre; Steenbock, Markus. A combination theorem for cubulation in small cancellation theory over free products. Annales de l'Institut Fourier, Tome 67 (2017) no. 4, pp. 1613-1670. doi : 10.5802/aif.3118. http://archive.numdam.org/item/AIF_2017__67_4_1613_0/

[1] Agol, Ian The virtual Haken conjecture, Doc. Math., Volume 18 (2013), pp. 1045-1087 (With an appendix by I. Agol, D. Groves, and J. Manning) | MR 3104553

[2] Arzhantseva, Goulnara; Steenbock, Markus Rips construction without unique product (2014) (https://arxiv.org/abs/1407.2441)

[3] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Springer, Berlin, 1999, xxii+643 pages | MR 1744486 (2000k:53038)

[4] Chatterji, Indira; Niblo, Graham From wall spaces to CAT (0) cube complexes, Internat. J. Algebra Comput., Volume 15 (2005) no. 5-6, pp. 875-885 | Article | MR 2197811 (2006m:20064)

[5] Corson, Jon Michael Complexes of groups, Proc. Lond. Math. Soc., Volume 65 (1992) no. 1, pp. 199-224 | Article | MR 1162493 (93h:57003)

[6] Edjvet, Martin; Juhász, Arye Nonsingular equations over groups II, Commun. Algebra, Volume 38 (2010) no. 5, pp. 1640-1657 | Article

[7] Edjvet, Martin; Juhász, Arye Non-singular equations over groups I, Algebra Colloq., Volume 18 (2011) no. 2, pp. 221-240 | Article

[8] Gerasimov, Victor N. Semi-splittings of groups and actions on cubings, Algebra, geometry, analysis and mathematical physics, Izdatel’stvo Instituta Matematiki, 1997, p. 91-109, 190 | MR 1624115 (99c:20049)

[9] Gruber, Dominik; Martin, Antonio; Steenbock, Markus Finite index subgroups without unique product in graphical small cancellation groups, Bull. Lond. Math. Soc., Volume 47 (2015) no. 4, pp. 631-638 | Article

[10] Gruber, Dominik; Sisto, Alessandro Infinitely presented graphical small cancellation groups are acylindrically hyperbolic (2014) (https://arxiv.org/abs/1408.4488v1)

[11] Haefliger, André Complexes of groups and orbihedra, Group theory from a geometrical viewpoint (Trieste, 1990), World Scientific, 1991, pp. 504-540 | MR 1170375 (93m:20048)

[12] Haglund, Frédéric; Paulin, Frédéric Simplicité de groupes d’automorphismes d’espaces à courbure négative, The Epstein birthday schrift (Geom. Topol. Monogr.) Volume 1, Geom. Topol. Publ., Coventry, 1998, p. 181-248 (electronic) | Article | MR 1668359 (2000b:20034)

[13] Haglund, Frédéric; Wise, Daniel T. Special cube complexes, Geom. Funct. Anal., Volume 17 (2008) no. 5, pp. 1551-1620 | Article

[14] Haglund, Frédéric; Wise, Daniel T. A combination theorem for special cube complexes. I, Ann. Math., Volume 176 (2012) no. 3, pp. 1427-1482 | Article

[15] Higson, Nigel; Kasparov, Gennadi E-theory and KK-theory for groups which act properly and isometrically on Hilbert space, Invent. Math., Volume 144 (2001) no. 1, pp. 23-74 | Article

[16] Hruska, G. Christopher; Wise, Daniel T. Finiteness properties of cubulated groups, Compos. Math., Volume 150 (2014) no. 3, pp. 453-506 | Article

[17] Hsu, Tim; Wise, Daniel T. Cubulating malnormal amalgams, Invent. Math., Volume 199 (2015) no. 2, pp. 293-331 | Article

[18] Linnell, Peter; Okun, Boris; Schick, Thomas The strong Atiyah conjecture for right-angled Artin and Coxeter groups, Geom. Dedicata, Volume 158 (2012), pp. 261-266 | Article

[19] Lyndon, Roger C.; Schupp, Paul E. Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 89, Springer, Berlin, 1977, xiv+339 pages | MR 1812024 (2001i:20064)

[20] Martin, Alexandre Non-positively curved complexes of groups and boundaries, Geom. Topol., Volume 18 (2014) no. 1, pp. 31-102 | Article

[21] Martin, Alexandre Combination of universal spaces for proper actions, J. Homotopy Relat. Struct., Volume 10 (2015) no. 4, pp. 803-820 | Article

[22] The Kourovka notebook. Unsolved Problems in Group Theory (Mazurov, Victor Danilovich; Khukro, Evgenii I., eds.), Institute of Mathematics, Russian Academy of Sciences, 2014 (https://arxiv.org/abs/1401.0300v3)

[23] McCammond, Jonathan P.; Wise, Daniel T. Fans and ladders in small cancellation theory, Proc. Lond. Math. Soc., Volume 84 (2002) no. 3, pp. 599-644 | Article | MR 1888425 (2003b:20047)

[24] Miller, Charles F. III; Schupp, Paul E. Embeddings into Hopfian groups, J. Algebra, Volume 17 (1971), pp. 171-176 | Article

[25] Nica, Bogdan Cubulating spaces with walls, Algebr. Geom. Topol., Volume 4 (2004), p. 297-309 (electronic) | Article

[26] Pankratʼev, A. E. Hyperbolic products of groups, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1999) no. 2, p. 9-13, 72

[27] Rips, Eliyahu; Segev, Yoav Torsion-free group without unique product property, J. Algebra, Volume 108 (1987) no. 1, pp. 116-126 | Article | MR 887195 (88g:20071)

[28] Sageev, Michah Ends of group pairs and non-positively curved cube complexes, Proc. Lond. Math. Soc., Volume 71 (1995) no. 3, pp. 585-617 | Article | MR 1347406 (97a:20062)

[29] Schreve, Kevin The strong Atiyah conjecture for virtually cocompact special groups, Math. Ann., Volume 359 (2014) no. 3-4, pp. 629-636 | Article

[30] Schupp, Paul E. Embeddings into simple groups, J. Lond. Math. Soc., Volume 13 (1976) no. 1, pp. 90-94 | Article

[31] Stallings, John R. Non-positively curved triangles of groups, Group theory from a geometrical viewpoint (Trieste, 1990), World Scientific, 1991, pp. 491-503 | MR 1170374 (94b:20033)

[32] Steenbock, Markus Rips–Segev torsion-free groups without the unique product property, J. Algebra, Volume 438 (2015), pp. 337-378 | Article

[33] Wise, Daniel T. Cubulating small cancellation groups, Geom. Funct. Anal., Volume 14 (2004) no. 1, pp. 150-214 | Article | MR 2053602 (2005c:20069)

[34] Wise, Daniel T. The Structure of Groups with a Quasiconvex Hierarchy (2011) (https://docs.google.com/file/d/0B45cNx80t5-2T0twUDFxVXRnQnc)