A combination theorem for cubulation in small cancellation theory over free products

Martin, Alexandre; Steenbock, Markus

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 $C^{\text{'}}(1/6)$ 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 $C^{\text{'}}(1/6)$–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.

Classification : 20F06, 20F65, 20F67
Keywords: group actions on CAT(0) cube complexes, small cancellation theory over free products, cubulation of groups.
Mot clés : actions de groupes sur des complexes cubiques CAT(0), théorie de la petite simplification sur un produit libre, cubulation de groupes.

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