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Proper isometric actions of hyperbolic groups on $L^p$-spaces

Part of: Connections with homological algebra and category theory Analysis on metric spaces Special aspects of infinite or finite groups Abstract harmonic analysis

Published online by Cambridge University Press:  26 February 2013

Bogdan Nica
Bogdan Nica*
Affiliation:
Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstrasse 3–5, D-37073 Göttingen, Germany (email: bogdan.nica@gmail.com)
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Abstract

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We show that every non-elementary hyperbolic group $\G $ admits a proper affine isometric action on $L^p(\bd \G \times \bd \G )$, where $\bd \G $ denotes the boundary of $\G $ and $p$ is large enough. Our construction involves a $\G $-invariant measure on $\bd \G \times \bd \G $ analogous to the Bowen–Margulis measure from the ${\rm CAT}(-1)$ setting, as well as a geometric, Busemann-type cocycle. We also deduce that $\G $ admits a proper affine isometric action on the first $\ell ^p$-cohomology group $H^1_{(p)}(\G )$ for large enough $p$.

Keywords

isometric actions on Banach spaceshyperbolic groupsBowen–Margulis measure$\ell ^p$-cohomology

MSC classification

Secondary: 20F67: Hyperbolic groups and nonpositively curved groups 20J06: Cohomology of groups 30L10: Quasiconformal mappings in metric spaces 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc.

Information

Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 5 , May 2013 , pp. 773 - 792
Copyright
Copyright © 2013 The Author(s) 

References

[Ada94]Adams, S., Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups, Topology 33 (1994), 765–783.Google Scholar
[BFGM07]Bader, U., Furman, A., Gelander, T. and Monod, N., Property (T) and rigidity for actions on Banach spaces, Acta Math. 198 (2007), 57–105.Google Scholar
[BlHV08]Bekka, B., de la Harpe, P. and Valette, A., Kazhdan’s property (T), New Mathematical Monographs, vol. 11 (Cambridge University Press, Cambridge, 2008).Google Scholar
[BHM11]Blachère, S., Haïssinsky, P. and Mathieu, P., Harmonic measures versus quasiconformal measures for hyperbolic groups, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), 683–721.Google Scholar
[Bou95]Bourdon, M., Structure conforme au bord et flot géodésique d’un ${\rm CAT}(-1)$-espace, Enseign. Math. (2) 41 (1995), 63–102.Google Scholar
[Bou]Bourdon, M., Cohomologie et actions isométriques propres sur les espaces $L_p$, in Geometry, topology and dynamics in negative curvature, Proceedings of the 2010 Bangalore Conference, to appear.Google Scholar
[BP03]Bourdon, M. and Pajot, H., Cohomologie $\ell _p$ et espaces de Besov, J. Reine Angew. Math. 558 (2003), 85–108.Google Scholar
[CCJJV01]Cherix, P.-A., Cowling, M., Jolissaint, P., Julg, P. and Valette, A., Groups with the Haagerup property (Gromov’s a-T-menability), Progress in Mathematics, vol. 197 (Birkhäuser, Basel, 2001).Google Scholar
[Coo93]Coornaert, M., Mesures de Patterson–Sullivan sur le bord d’un espace hyperbolique au sens de Gromov, Pacific J. Math. 159 (1993), 241–270.Google Scholar
[EN12]Emerson, H. and Nica, B., Finitely summable Fredholm modules for boundary actions of hyperbolic groups, Preprint (2012), arXiv:1208.0856.Google Scholar
[FP82]Figà-Talamanca, A. and Picardello, M. A., Spherical functions and harmonic analysis on free groups, J. Funct. Anal. 47 (1982), 281–304.Google Scholar
[Fur02]Furman, A., Coarse-geometric perspective on negatively curved manifolds and groups, in Rigidity in dynamics and geometry (Springer, 2002), 149–166.Google Scholar
[KB02]Kapovich, I. and Benakli, N., Boundaries of hyperbolic groups, in Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), Contemporary Mathematics, vol. 296 (American Mathematical Society, Providence, RI, 2002), 39–93.Google Scholar
[Min05]Mineyev, I., Flows and joins of metric spaces, Geom. Topol. 9 (2005), 403–482.Google Scholar
[Min07]Mineyev, I., Metric conformal structures and hyperbolic dimension, Conform. Geom. Dyn. 11 (2007), 137–163.Google Scholar
[MY02]Mineyev, I. and Yu, G., The Baum–Connes conjecture for hyperbolic groups, Invent. Math. 149 (2002), 97–122.Google Scholar
[Nic12]Nica, B., The Mazur–Ulam theorem, Expo. Math. 30 (2012), 397–398.Google Scholar
[Pan89]Pansu, P., Dimension conforme et sphère à l’infini des variétés à courbure négative, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), 177–212.Google Scholar
[Rud87]Rudin, W., Real and complex analysis, third edition (McGraw-Hill, New York, NY, 1987).Google Scholar
[Sul79]Sullivan, D., The density at infinity of a discrete group of hyperbolic motions, Publ. Inst. Hautes Études Sci. 50 (1979), 171–202.Google Scholar
[Yu05]Yu, G., Hyperbolic groups admit proper affine isometric actions on $\ell ^p$-spaces, Geom. Funct. Anal. 15 (2005), 1144–1151.Google Scholar