We show that superreflexivity can be characterized in terms of bilipschitz embeddability of word hyperbolic groups.We compare characterizations of superrefiexivity in terms of diamond graphs and binary trees.We show that there exist sequences of series-parallel graphs of increasing topological complexitywhich admit uniformly bilipschitz embeddings into a Hilbert space, and thus do not characterize superrefiexivity.
@article{bwmeta1.element.doi-10_2478_agms-2014-0005,
author = {Mikhail Ostrovskii},
title = {Metric Characterizations of Superreflexivity in Terms of Word Hyperbolic Groups and Finite Graphs},
journal = {Analysis and Geometry in Metric Spaces},
volume = {2},
year = {2014},
zbl = {1318.46010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_agms-2014-0005}
}
Mikhail Ostrovskii. Metric Characterizations of Superreflexivity in Terms of Word Hyperbolic Groups and Finite Graphs. Analysis and Geometry in Metric Spaces, Tome 2 (2014) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_agms-2014-0005/
[1] G. N. Arzhantseva, On quasiconvex subgroups of word hyperbolic groups, Geom. Dedicata, 87 (2001), no. 1-3, 191-208. | Zbl 0994.20036
[2] F. Baudier, Metrical characterization of super-reflexivity and linear type of Banach spaces, Archiv Math., 89 (2007), no. 5, 419-429.[WoS] | Zbl 1142.46007
[3] B. Beauzamy, Introduction to Banach spaces and their geometry. North-Holland Mathematics Studies, 68. Notas de Matemática [Mathematical Notes], 86. North-Holland Publishing Co., Amsterdam-New York, 1982. Second Edition: 1985.
[4] I. Benjamini, O. Schramm, Every graph with a positive Cheeger constant contains a tree with a positive Cheeger constant, Geom. Funct. Anal. 7 (1997), no. 3, 403-419. | Zbl 0882.05052
[5] Y. Benyamini, J. Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1. AmericanMathematical Society Colloquium Publications, 48. American Mathematical Society, Providence, RI, 2000. | Zbl 0946.46002
[6] J. Bourgain, The metrical interpretation of superreflexivity in Banach spaces, Israel J. Math., 56 (1986), no. 2, 222-230. | Zbl 0643.46013
[7] M. R. Bridson, A. Haefliger, Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319. Springer-Verlag, Berlin, 1999.
[8] B. Brinkman, A. Karagiozova, J. R. Lee, Vertex cuts, random walks, and dimension reduction in series-parallel graphs, in: STOC’07-Proceedings of the 39th Annual ACM Symposium on Theory of Computing, 621-630, ACM, New York, 2007. | Zbl 1232.68163
[9] S. Buyalo, A. Dranishnikov, V. Schroeder, Embedding of hyperbolic groups into products of binary trees, Invent. Math., 169 (2007), no. 1, 153-192. | Zbl 1157.57003
[10] S. Buyalo, V. Schroeder, Elements of asymptotic geometry, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2007. | Zbl 1125.53036
[11] F. Dahmani, V. Guirardel, D. Osin, Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, arXiv:1111.7048v3.
[12] R. Deville, G. Godefroy, V. Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, 64, Longman Scienti_c & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. | Zbl 0782.46019
[13] M. Deza, M. Laurent, Geometry of cuts and metrics. Algorithms and Combinatorics, 15. Springer-Verlag, Berlin, 1997. | Zbl 0885.52001
[14] D. van Dulst, Reflexive and superreflexive Banach spaces. Mathematical Centre Tracts, 102. Mathematisch Centrum, Amsterdam, 1978. | Zbl 0412.46006
[15] J. Elton, E. Odell, The unit ball of every in_nite-dimensional normed linear space contains a (1 + ")-separated sequence. Colloq. Math. 44 (1981), no. 1, 105-109. | Zbl 0493.46014
[16] P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math, 13 (1972), 281-288.
[17] D. Eppstein, Parallel recognition of series-parallel graphs. Inform. and Comput. 98 (1992), no. 1, 41-55. | Zbl 0754.68056
[18] M. Gromov, Hyperbolic groups, in: Essays in group theory, 75-263,Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987.; Russian translation: Institute of Computer Science, Izhevsk, 2002.
[19] A. Gupta, I. Newman, Y. Rabinovich, A. Sinclair, Cuts, trees and `1-embeddings of graphs, Combinatorica, 24 (2004) 233-269; Conference version in: 40th Annual IEEE Symposium on Foundations of Computer Science, 1999, pp. 399-408. | Zbl 1056.05040
[20] R. C. James, Uniformly non-square Banach spaces. Ann. of Math. (2) 80 (1964), 542-550. | Zbl 0132.08902
[21] R. C. James, Some self-dual properties of normed linear spaces, in: Symposiumon In_nite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967), pp. 159-175. Ann. ofMath. Studies, No. 69, Princeton Univ. Press, Princeton, N. J., 1972.
[22] R. C. James, Super-reflexive Banach spaces, Canad. J. Math., 24 (1972), 896-904. | Zbl 0222.46009
[23] W. B. Johnson, G. Schechtman, Diamond graphs and super-reflexivity, J. Topol. Anal., 1 (2009), no. 2, 177-189.[WoS][Crossref] | Zbl 1183.46022
[24] B. Kloeckner, Yet another short proof of the Bourgain’s distortion estimate for embedding of trees into uniformly convex Banach spaces, Israel J. Math., to appear, DOI: 10.1007/s11856-014-0024-4, http://www-fourier.ujfgrenoble.fr/_bkloeckn/recherche.html[Crossref] | Zbl 1314.46028
[25] C. A. Kottman, Subsets of the unit ball that are separated by more than one. Studia Math. 53 (1975), no. 1, 15-27. | Zbl 0266.46014
[26] M. Mendel, A. Naor, Markov convexity and local rigidity of distorted metrics, J. Eur. Math. Soc. (JEMS), 15 (2013), no. 1, 287-337; Conference version: Computational geometry (SCG’08), 49-58, ACM, New York, 2008.[Crossref] | Zbl 1266.46016
[27] P.W. Nowak, G. Yu, Large scale geometry. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2012. | Zbl 1264.53051
[28] M. I. Ostrovskii, Embeddability of locally _nite metric spaces into Banach spaces is _nitely determined, Proc. Amer. Math. Soc., 140 (2012), 2721-2730. [29] M. I. Ostrovskii, Metric Embeddings: Bilipschitz and Coarse Embeddings into Banach Spaces, de Gruyter Studies in Mathematics, 49. Walter de Gruyter & Co., Berlin, 2013. | Zbl 1276.46013
[30] M. I. Ostrovskii, Test-space characterizations of some classes of Banach spaces, in: Algebraic Methods in Functional Analysis, The Victor Shulman Anniversary Volume, I. G. Todorov, L. Turowska (Eds.), Operator Theory: Advances and Applications, Vol. 233, Birkhäuser, Basel, 2013, pp. 103-126.
[31] G. Pisier, Martingales in Banach spaces (in connection with type and cotype), Lecture notes of a course given at l’Institut Henri Poincaré, February 2-8, 2011, 242 pp; see the web site: http://perso-math.univ-mlv.fr/users/banach/Winterschool2011/
[32] Y. Rabinovich, R. Raz, Lower bounds on the distortion of embedding _nite metric spaces in graphs, Discrete Comput. Geom., 19 (1998), no. 1, 79-94. | Zbl 0890.05021
[33] J. J. Schä_er, K. Sundaresan, Reflexivity and the girth of spheres. Math. Ann. 184 (1969/1970) 163-168.
[34] A. Sisto, Quasi-convexity of hyperbolically embedded subgroups, Math. Z. to appear, arXiv: 1310.7753.