Metric Characterizations of Superreflexivity in Terms of Word Hyperbolic Groups and Finite Graphs

Mikhail Ostrovskii

Analysis and Geometry in Metric Spaces (2014)

  • Volume: 2, Issue: 1, page 154-168, electronic only
  • ISSN: 2299-3274

Abstract

top
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.

How to cite

top

Mikhail Ostrovskii. "Metric Characterizations of Superreflexivity in Terms of Word Hyperbolic Groups and Finite Graphs." Analysis and Geometry in Metric Spaces 2.1 (2014): 154-168, electronic only. <http://eudml.org/doc/267138>.

@article{MikhailOstrovskii2014,
abstract = {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.},
author = {Mikhail Ostrovskii},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {bi-Lipschitz embedding; diamond graphs; series-parallel graph; superreflexivity;word hyperbolic group; superreflexivity; word hyperbolic group},
language = {eng},
number = {1},
pages = {154-168, electronic only},
title = {Metric Characterizations of Superreflexivity in Terms of Word Hyperbolic Groups and Finite Graphs},
url = {http://eudml.org/doc/267138},
volume = {2},
year = {2014},
}

TY - JOUR
AU - Mikhail Ostrovskii
TI - Metric Characterizations of Superreflexivity in Terms of Word Hyperbolic Groups and Finite Graphs
JO - Analysis and Geometry in Metric Spaces
PY - 2014
VL - 2
IS - 1
SP - 154
EP - 168, electronic only
AB - 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.
LA - eng
KW - bi-Lipschitz embedding; diamond graphs; series-parallel graph; superreflexivity;word hyperbolic group; superreflexivity; word hyperbolic group
UR - http://eudml.org/doc/267138
ER -

References

top
  1. [1] G. N. Arzhantseva, On quasiconvex subgroups of word hyperbolic groups, Geom. Dedicata, 87 (2001), no. 1-3, 191-208. Zbl0994.20036
  2. [2] F. Baudier, Metrical characterization of super-reflexivity and linear type of Banach spaces, Archiv Math., 89 (2007), no. 5, 419-429.[WoS] Zbl1142.46007
  3. [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. [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. Zbl0882.05052
  5. [5] Y. Benyamini, J. Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1. AmericanMathematical Society Colloquium Publications, 48. American Mathematical Society, Providence, RI, 2000. Zbl0946.46002
  6. [6] J. Bourgain, The metrical interpretation of superreflexivity in Banach spaces, Israel J. Math., 56 (1986), no. 2, 222-230. Zbl0643.46013
  7. [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. [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. Zbl1232.68163
  9. [9] S. Buyalo, A. Dranishnikov, V. Schroeder, Embedding of hyperbolic groups into products of binary trees, Invent. Math., 169 (2007), no. 1, 153-192. Zbl1157.57003
  10. [10] S. Buyalo, V. Schroeder, Elements of asymptotic geometry, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2007. Zbl1125.53036
  11. [11] F. Dahmani, V. Guirardel, D. Osin, Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, arXiv:1111.7048v3. 
  12. [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. Zbl0782.46019
  13. [13] M. Deza, M. Laurent, Geometry of cuts and metrics. Algorithms and Combinatorics, 15. Springer-Verlag, Berlin, 1997. Zbl0885.52001
  14. [14] D. van Dulst, Reflexive and superreflexive Banach spaces. Mathematical Centre Tracts, 102. Mathematisch Centrum, Amsterdam, 1978. Zbl0412.46006
  15. [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. Zbl0493.46014
  16. [16] P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math, 13 (1972), 281-288. 
  17. [17] D. Eppstein, Parallel recognition of series-parallel graphs. Inform. and Comput. 98 (1992), no. 1, 41-55. Zbl0754.68056
  18. [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. [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. Zbl1056.05040
  20. [20] R. C. James, Uniformly non-square Banach spaces. Ann. of Math. (2) 80 (1964), 542-550. Zbl0132.08902
  21. [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. [22] R. C. James, Super-reflexive Banach spaces, Canad. J. Math., 24 (1972), 896-904. Zbl0222.46009
  23. [23] W. B. Johnson, G. Schechtman, Diamond graphs and super-reflexivity, J. Topol. Anal., 1 (2009), no. 2, 177-189.[WoS][Crossref] Zbl1183.46022
  24. [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] Zbl1314.46028
  25. [25] C. A. Kottman, Subsets of the unit ball that are separated by more than one. Studia Math. 53 (1975), no. 1, 15-27. Zbl0266.46014
  26. [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] Zbl1266.46016
  27. [27] P.W. Nowak, G. Yu, Large scale geometry. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2012. Zbl1264.53051
  28. [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. Zbl1276.46013
  29. [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. 
  30. [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/ 
  31. [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. Zbl0890.05021
  32. [33] J. J. Schä_er, K. Sundaresan, Reflexivity and the girth of spheres. Math. Ann. 184 (1969/1970) 163-168. 
  33. [34] A. Sisto, Quasi-convexity of hyperbolically embedded subgroups, Math. Z. to appear, arXiv: 1310.7753. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.