Proper actions and weak amenability of classical relatively hyperbolic groups
[Actions propres et moyennabilité faible des groupes relativement hyperboliques classiques]
Guentner, Erik1 ; Reckwerdt, Eric2, 3 ; Tessera, Romain4
1 Department of Mathematics University of Hawai‘i at Mānoa Honolulu, HI (USA)
2 Institute of Mathematics Polish Academy of Sciences Warsaw (Poland)
3 Current address: Computer Science Department University of Colorado Boulder, CO (USA)
4 Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université de Paris, Paris (France)
Annales de l'Institut Fourier, Online first, 31 p.

Gromov introduced a notion of hyperbolicity for discrete groups (and general metric spaces) as an abstraction of the properties of universal covers of closed, negatively curved manifolds and their fundamental groups. The fundamental group of a manifold with pinched negative curvature and a cusp is not hyperbolic, but it is relatively hyperbolic with respect to the cusp subgroup, which has polynomial growth. We introduce a thinning technique which allows to reduce questions about these classical relatively hyperbolic groups to the case of bounded geometry hyperbolic graphs. As applications, we show that such groups admit a proper affine action on an L p -space and are weakly amenable in the sense of Cowling–Haagerup. These results generalize earlier work of G. Yu and N. Ozawa, respectively, from the setting of hyperbolic groups to classical relatively hyperbolic groups.

Gromov a introduit la notion d’hyperbolicité pour les groupes discrets (et les espaces métriques généraux) comme une abstraction des propriétés métriques des revêtement universels de variétés compactes à courbure sectionnelle strictement négative, et de leurs groupes fondamentaux. Le groupe fondamental d’une variété à courbure négative pincée possédant un cusp n’est pas hyperbolique, mais est relativement hyperbolique par rapport au groupe de cusp, lequel est à croissance polynomiale. Nous introduisons une technique d’affinage permettant de ramener l’étude de ces groupes relativement hyperboliques “classiques” à celle de graphes hyperboliques de degré borné. Comme applications, nous démontrons que de tels groupes admettent une action propre par isométries affines sur un espace L p , et sont faiblement moyennables au sens de Cowling–Haagerup. Ces résultats généralisent respectivement les travaux de G. Yu et N. Ozawa, qui avaient démontré ces propriétés pour les groupes hyperboliques.

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DOI : 10.5802/aif.3737
Classification : 22D55, 20F65
Keywords: Relatively hyperbolic group, Haagerup property, weak amenability
Mots-clés : Groupes relativement hyperboliques, Propriété de Haagerup, moyennabilité faible

Guentner, Erik 1 ; Reckwerdt, Eric 2, 3 ; Tessera, Romain 4

1 Department of Mathematics University of Hawai‘i at Mānoa Honolulu, HI (USA)
2 Institute of Mathematics Polish Academy of Sciences Warsaw (Poland)
3 Current address: Computer Science Department University of Colorado Boulder, CO (USA)
4 Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université de Paris, Paris (France) 
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Guentner, Erik; Reckwerdt, Eric; Tessera, Romain. Proper actions and weak amenability of classical relatively hyperbolic groups. Annales de l'Institut Fourier, Online first, 31 p.

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