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Poincaré inequalities, embeddings, and wild groups

Part of: Special aspects of infinite or finite groups Normed linear spaces and Banach spaces; Banach lattices Variational problems in infinite-dimensional spaces

Published online by Cambridge University Press:  24 August 2011

Assaf Naor  and
Lior Silberman
Assaf Naor
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY, USA (email: naor@cims.nyu.edu)
Lior Silberman
Affiliation:
Department of Mathematics, The University of British Columbia, Vancouver, BC, Canada (email: lior@math.ubc.ca)
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Abstract

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We present geometric conditions on a metric space (Y,dY) ensuring that, almost surely, any isometric action on Y by Gromov’s expander-based random group has a common fixed point. These geometric conditions involve uniform convexity and the validity of nonlinear Poincaré inequalities, and they are stable under natural operations such as scaling, Gromov–Hausdorff limits, and Cartesian products. We use methods from metric embedding theory to establish the validity of these conditions for a variety of classes of metric spaces, thus establishing new fixed point results for actions of Gromov’s ‘wild groups’.

Keywords

Gromov’s random groupsfixed pointsPoincaré inequalities

MSC classification

Secondary: 20F65: Geometric group theory 58E40: Group actions 46B85: Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 5 , September 2011 , pp. 1546 - 1572
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[AS00]Alon, N. and Spencer, J. H., The probabilistic method, Wiley-Interscience Series in Discrete Mathematics and Optimization, second edition (John Wiley & Sons, New York, 2000), with an appendix on the life and work of Paul Erdős; MR 1885388(2003f:60003).CrossRefGoogle Scholar
[ABJLMS09]Arzhantseva, G., Bridson, M. R., Januszkiewicz, T., Leary, I. J., Minasyan, A. and Świa̧tkowski, J., Infinite groups with fixed point properties, Geom. Topol. 13 (2009), 1229–1263; MR 2496045(2010b:20069).CrossRefGoogle Scholar
[AD08]Arzhantseva, G. A. and Delzant, T., Examples of random groups, Preprint (2008), submitted, available at http://www.unige.ch/math/folks/arjantse/publicationsGA.html.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; MR 2316269(2008g:22007).Google Scholar
[BCL94]Ball, K., Carlen, E. A. and Lieb, E. H., Sharp uniform convexity and smoothness inequalities for trace norms, Invent. Math. 115 (1994), 463–482; MR 1262940(95e:47027).Google Scholar
[BLMN05]Bartal, Y., Linial, N., Mendel, M. and Naor, A., On metric Ramsey type phenomena, Ann. of Math. (2) 162 (2005), 643–709.CrossRefGoogle Scholar
[Bou85]Bourgain, J., On Lipschitz embedding of finite metric spaces in Hilbert space, Israel J. Math. 52 (1985), 46–52; MR 815600(87b:46017).Google Scholar
[BG07]Breuillard, E. and Gelander, T., A topological Tits alternative, Ann. of Math. (2) 166 (2007), 427–474; MR 2373146(2009a:20077).Google Scholar
[BH99]Bridson, M. R. and Haefliger, A., Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319 (Springer, Berlin, 1999); MR 2000k:53038.CrossRefGoogle Scholar
[Del77]Delorme, P., 1-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations, Bull. Soc. Math. France 105 (1977), 281–336; MR 58#28272.CrossRefGoogle Scholar
[Fig76]Figiel, T., On the moduli of convexity and smoothness, Studia Math. 56 (1976), 121–155; MR 0425581(54#13535).CrossRefGoogle Scholar
[FM05]Fisher, D. and Margulis, G., Almost isometric actions, property (T), and local rigidity, Invent. Math. 162 (2005), 19–80.CrossRefGoogle Scholar
[FS08]Fisher, D. and Silberman, L., Groups not acting on manifolds, Int. Math. Res. Not. IMRN (2008), 11, Art. ID rnn060; MR 2435750(2009f:57059).Google ScholarPubMed
[Gro03]Gromov, M., Random walk in random groups, Geom. Funct. Anal. 13 (2003), 73–146.CrossRefGoogle Scholar
[GHW05]Guentner, E., Higson, N. and Weinberger, S., The Novikov conjecture for linear groups, Publ. Math. Inst. Hautes Études Sci. (2005), 243–268; MR 2217050(2007c:19007).Google Scholar
[Gui77]Guichardet, A., Étude de la l-cohomologie et de la topologie du dual pour les groupes de Lie à radical abélien, Math. Ann. 228 (1977), 215–232; MR 56#539.CrossRefGoogle Scholar
[HLS02]Higson, N., Lafforgue, V. and Skandalis, G., Counterexamples to the Baum–Connes conjecture, Geom. Funct. Anal. 12 (2002), 330–354; MR 1911663(2003g:19007).Google Scholar
[HLW06]Hoory, S., Linial, N. and Wigderson, A., Expander graphs and their applications, Bull. Amer. Math. Soc. (N.S.) 43 (2006), 439–561 (electronic); MR 2247919(2007h:68055).CrossRefGoogle Scholar
[IKN09]Izeki, H., Kondo, T. and Nayatani, S., Fixed-point property of random groups, Ann. Global Anal. Geom. 35 (2009), 363–379; MR 2506240.Google Scholar
[IN05]Izeki, H. and Nayatani, S., Combinatorial harmonic maps and discrete-group actions on Hadamard spaces, Geom. Dedicata 114 (2005), 147–188; MR 2174098(2006k:58024).CrossRefGoogle Scholar
[Jos97]Jost, J., Nonpositive curvature: geometric and analytic aspects, Lectures in Mathematics ETH Zürich (Birkhäuser, Basel, 1997); MR 1451625(98g:53070).Google Scholar
[Kap05]Kapovich, M., Representations of polygons of finite groups, Geom. Topol. 9 (2005), 1915–1951 (electronic); MR 2175160(2006g:20069).CrossRefGoogle Scholar
[KL97]Kleiner, B. and Leeb, B., Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Publ. Math. Inst. Hautes Études Sci. 86 (1997), 115–197 (1998); MR 1608566(98m:53068).Google Scholar
[LS05]Lang, U. and Schlichenmaier, T., Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions, Int. Math. Res. Not. IMRN (2005), 3625–3655; MR 2200122.Google Scholar
[LMN05]Lee, J. R., Mendel, M. and Naor, A., Metric structures in L 1: dimension, snowflakes, and average distortion, European. J. Combin. 26 (2005), 1180–1190.Google Scholar
[LN05]Lee, J. R. and Naor, A., Extending Lipschitz functions via random metric partitions, Invent. Math. 160 (2005), 59–95.Google Scholar
[Mat97]Matoušek, J., On embedding expanders into l p spaces, Israel J. Math. 102 (1997), 189–197; MR 1489105(98k:46014).Google Scholar
[Nag83]Nagata, J.-i., Modern dimension theory, Sigma Series in Pure Mathematics, vol. 2, revised edition (Heldermann, Berlin, 1983); MR 715431(84h:54033).Google Scholar
[NPSS06]Naor, A., Peres, Y., Schramm, O. and Sheffield, S., Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces, Duke Math. J. 134 (2006), 165–197; MR 2239346(2007k:46017).Google Scholar
[NR05]Naor, A. and Rabani, Y., Spectral inequalities on curved spaces, Preprint (2005).Google Scholar
[Oll05]Ollivier, Y., A January 2005 invitation to random groups, Ensaios Matemáticos [Mathematical Surveys], vol. 10 (Sociedade Brasileira de Matemática, Rio de Janeiro, 2005), available at http://www.umpa.ens-lyon.fr/∼yollivie/publs.html; MR 220536.Google Scholar
[Oll06]Ollivier, Y., On a small cancellation theorem of Gromov, Bull. Belg. Math. Soc. Simon Stevin 13 (2006), 75–89; MR 2245980.CrossRefGoogle Scholar
[OW07]Ollivier, Y. and Wise, D. T., Kazhdan groups with infinite outer automorphism group, Trans. Amer. Math. Soc. 359 (2007), 1959–1976 (electronic); MR 2276608(2008a:20049).CrossRefGoogle Scholar
[Oza04]Ozawa, N., A note on non-amenability of ℬ(l p) for p=1,2, Internat. J. Math. 15 (2004), 557–565; MR 2078880(2005g:46135).Google Scholar
[Pan95]Pansu, P., Cohomologie L p: invariance sous quasiisométrie, Preprint (1995), available at http://www.math.u-psud.fr/∼pansu/qi04.pdf].Google Scholar
[Pan06]Pansu, P., Fixed points of actions of building groups on CAT(0) spaces, in Workshop ‘Group theory: geometric and probabilistic methods’, Ein Gedi, October 2006, http://www.math.u-psud.fr/∼pansu/ein_gedi_beamer.pdf.Google Scholar
[Sil03]Silberman, L., Addendum to: ‘Random walk in random groups’ by M. Gromov, Geom. Funct. Anal. 13 (2003), 147–177; MR 1978493(2004j:20088b).CrossRefGoogle Scholar
[Wan98]Wang, M.-T., A fixed point theorem of discrete group actions on Riemannian manifolds, J. Differential Geom. 50 (1998), 249–267; MR 1684980(2000e:53051).CrossRefGoogle Scholar
[Yu05]Yu, G., Hyperbolic groups admit proper affine isometric actions on l p-spaces, Geom. Funct. Anal. 15 (2005), 1144–1151; MR 2221161(2007f:20075).Google Scholar
[Zuk03]Żuk, A., Property (T) and Kazhdan constants for discrete groups, Geom. Funct. Anal. 13 (2003), 643–670; MR 1995802(2004m:20079).Google Scholar