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\(L_p\) compression, traveling salesmen, and stable walks. (English) Zbl 1268.20044

Summary: We show that if \(H\) is a group of polynomial growth whose growth rate is at least quadratic, then the \(L_p\) compression of the wreath product \(\mathbb Z\wr H\) equals \(\max\{\tfrac 1p,\tfrac 12\}\). We also show that the \(L_p\) compression of \(\mathbb Z\wr\mathbb Z\) equals \(\max\{\tfrac p{2p-1},\tfrac 23\}\) and that the \(L_p\) compression of \((\mathbb Z\wr\mathbb Z)_0\) (the zero section of \(\mathbb Z\wr\mathbb Z\), equipped with the metric induced from \(\mathbb Z\wr\mathbb Z\)) equals \(\max\{\tfrac{p+1}{2p},\tfrac 34\}\). The fact that the Hilbert compression exponent of \(\mathbb Z\wr\mathbb Z\) equals \(2/3\) while the Hilbert compression exponent of \((\mathbb Z\wr\mathbb Z)_0\) equals \(3/4\) is used to show that there exists a Lipschitz function \(f\colon(\mathbb Z\wr\mathbb Z)_0\to L_2\) which cannot be extended to a Lipschitz function defined on all of \(\mathbb Z\wr\mathbb Z\).

MSC:

20F65 Geometric group theory
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A07 Means on groups, semigroups, etc.; amenable groups
60G50 Sums of independent random variables; random walks
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[1] I. Aharoni, B. Maurey, and B. S. Mityagin, Uniform embeddings of metric spaces and of Banach spaces into Hilbert spaces , Israel J. Math. 52 (1985), 251-265. · Zbl 0596.46010 · doi:10.1007/BF02786521
[2] N. Alon and J. H. Spencer, The Probabilistic Method , 2nd. ed., Wiley-Intersci. Ser. Discrete Math. Optim., Wiley-Interscience, New York, 2000. · Zbl 0996.05001
[3] G. Arzhantseva, C. Druţu, and M. Sapir, Compression functions of uniform embeddings of groups into Hilbert and Banach spaces , J. Reine Angew. Math. 633 (2009), 213-235. · Zbl 1258.20032 · doi:10.1515/CRELLE.2009.066
[4] G. N. Arzhantseva, V. S. Guba, and M. V. Sapir, Metrics on diagram groups and uniform embeddings in a Hilbert space , Comment. Math. Helv. 81 (2006), 911-929. · Zbl 1166.20031 · doi:10.4171/CMH/80
[5] P. Assouad, Plongements Lipschitziens dans R \(^{n}\), Bull. Soc. Math. France 111 (1983), 429-448. · Zbl 0597.54015
[6] T. Austin, A finitely generated amenable group with very poor compression into Lebesgue spaces , · Zbl 1226.43001
[7] T. Austin, A. Naor, and Y. Peres, The wreath product of \(\mathbb Z\) with \(\mathbb Z\) has Hilbert compression exponent \(\frac{2}{3}\), Proc. Amer. Math. Soc. 137 (2009), 85-90. · Zbl 1226.20032 · doi:10.1090/S0002-9939-08-09501-4
[8] T. Austin, A. Naor, and A. Valette, The Euclidean distortion of the lamplighter group , Discrete Comput. Geom. 44 (2010), 55-74. · Zbl 1275.20044
[9] K. Ball, Markov chains, Riesz transforms and Lipschitz maps , Geom. Funct. Anal. 2 (1992), 137-172. · Zbl 0788.46050 · doi:10.1007/BF01896971
[10] Y. Bartal, N. Linial, M. Mendel, and A. Naor, On metric Ramsey-type phenomena , Ann. of Math. (2) 162 (2005), 643-709. · Zbl 1114.46007 · doi:10.4007/annals.2005.162.643
[11] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Vol. 1 , Amer. Math. Soc. Colloq. Publ. 48 , Amer. Math. Soc., Providence, 2000. · Zbl 0946.46002
[12] J. Bourgain, The metrical interpretation of superreflexivity in Banach spaces , Israel J. Math. 56 (1986), 222-230. · Zbl 0643.46013 · doi:10.1007/BF02766125
[13] J. Bretagnolle, D. Dacunha-Castelle, and J.-L. Krivine, Fonctions de type positif sur les espaces \(L^{p}\) , C. R. Math. Acad. Sci. Paris 261 (1965), 2153-2156. · Zbl 0255.42022
[14] N. Brodskiy and D. Sonkin, Compression of uniform embeddings into Hilbert space , Topology Appl. 155 (2008), 725-732. · Zbl 1191.20041 · doi:10.1016/j.topol.2007.12.012
[15] S. Campbell and G. A. Niblo, Hilbert space compression and exactness of discrete groups , J. Funct. Anal. 222 (2005), 292-305. · Zbl 1121.20032 · doi:10.1016/j.jfa.2005.01.012
[16] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces , Geom. Funct. Anal. 9 (1999), 428-517. · Zbl 0942.58018 · doi:10.1007/s000390050094
[17] J. Cheeger and B. Kleiner, Differentiating maps into \({L}^1\) and the geometry of BV functions , Ann. of Math. (2) 171 (2010), 1347-1385. · Zbl 1194.22009 · doi:10.4007/annals.2010.171.1347
[18] P.-A. Cherix, M. Cowling, P. Jolissaint, P. Julg, and A. Valette, Groups with the Haagerup Property , Progr. Math. 97 , Birkhäuser, Basel, 2001. · Zbl 1030.43002
[19] T. H. Colding and W. P. Minicozzi, Ii, Liouville theorems for harmonic sections and applications , Comm. Pure Appl. Math. 51 (1998), 113-138. · Zbl 0928.58022 · doi:10.1002/(SICI)1097-0312(199802)51:2<113::AID-CPA1>3.0.CO;2-E
[20] D. Dacunha-Castelle and J.-L. Krivine, Ultraproduits d’espaces d’Orlicz et applications géométriques , C. R. Math. Acad. Sci. Paris Sér. A-B 271 (1970), A987-A989. · Zbl 0213.12901
[21] -, Applications des ultraproduits à l’étude des espaces et des algèbres de Banach , Studia Math. 41 (1972), 315-334. · Zbl 0275.46023
[22] Y. De Cornulier, Y. Stalder, and A. Valette, Proper actions of lamplighter groups associated with free groups , C. R. Math. Acad. Sci. Paris 346 (2008), 173-176. · Zbl 1132.43001 · doi:10.1016/j.crma.2007.11.027
[23] -, Proper actions of wreath products and generalizations , · Zbl 1283.20049
[24] Y. De Cornulier, R. Tessera, and A. Valette, Isometric group actions on Hilbert spaces: Growth of cocycles , Geom. Funct. Anal. 17 (2007), 770-792. · Zbl 1129.22004 · doi:10.1007/s00039-007-0604-0
[25] P. De La Harpe and A. Valette, La propriété \((T)\) de Kazhdan pour les groupes localement compacts, avec un appendice de M. Burger, Astérisque 175 , Soc. Math. France, Paris, 1989. · Zbl 0759.22001
[26] N. Dunford and J. T. Schwartz, Linear Operators, Part I , Wiley Classics Lib., Wiley, New York, 1988. · Zbl 0635.47001
[27] R. Durrett, Probability: Theory and Examples , 2nd ed., Duxbury, Belmont, Calif, 1996. · Zbl 1202.60001
[28] A. G. èrschler [Anna Erschler], On the asymptotics of the rate of departure to infinity (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 283 (2001), 251-257; English translation in J. Math. Sci. (N.Y.) 121 , no. 3 (2004), 2437-2440. · Zbl 0993.70006
[29] W. Feller, An Introduction to Probability Theory and its Applications, Vol. II , Wiley, New York, 1966. · Zbl 0138.10207
[30] R. J. Fleming and J. E. Jamison, Isometries on Banach Spaces: Function Spaces , Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, Fla, 2003.
[31] ś. R. Gal, Asymptotic dimension and uniform embeddings , Groups Geom. Dyn. 2 (2008), 63-84. · Zbl 1192.20028 · doi:10.4171/GGD/31
[32] M. Gromov, Random walk in random groups , Geom. Funct. Anal. 13 (2003), 73-146. · Zbl 1122.20021 · doi:10.1007/s000390300002
[33] E. Guentner and J. Kaminker, Exactness and uniform embeddability of discrete groups , J. Lond. Math. Soc. (2) 70 (2004), 703-718. · Zbl 1082.46049 · doi:10.1112/S0024610704005897
[34] P. R. Halmos, Measure Theory , D. Van Nostrand, New York, 1950. · Zbl 0040.16802
[35] J. Heinonen, Lectures on Analysis on Metric Spaces , Universitext, Springer, New York, 2001. · Zbl 0985.46008 · doi:10.1007/978-1-4613-0131-8
[36] S. Heinrich, Ultraproducts in Banach space theory , J. Reine Angew. Math. 313 (1980), 72-104. · Zbl 0412.46017 · doi:10.1515/crll.1980.313.72
[37] I. A. Ibragimov and Y. V. Linnik, Independent and Stationary Sequences of Random Variables , Wolters-Noordhoff, Groningen, 1971. · Zbl 0219.60027
[38] P. W. Jones, Rectifiable sets and the traveling salesman problem , Invent. Math. 102 (1990), 1-15. · Zbl 0731.30018 · doi:10.1007/BF01233418
[39] S. Kakutani, Concrete representation of abstract \((L)\) -spaces and the mean ergodic theorem, Ann. of Math. (2) 42 (1941), 523-537. JSTOR: · Zbl 0027.11102 · doi:10.2307/1968915
[40] D. A. Každan, Connection of the dual space of a group with the structure of its closed subgroups (in Russian), Funkcional. Anal. i Priložen. 1 , no. 1 (1967), 71-74; English translation in Funct. Anal. Appl. 1 , no. 1 (1967), 63-65.
[41] J. Lamperti, On the isometries of certain function-spaces , Pacific J. Math. 8 (1958), 459-466. · Zbl 0085.09702 · doi:10.2140/pjm.1958.8.459
[42] S. Li, Compression bounds for wreath products , Proc. Amer. Math. Soc. 138 (2010), 2701-2714. · Zbl 1235.20042 · doi:10.1090/S0002-9939-10-10307-4
[43] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, II , Ergeb. Math. Grenzgeb. (3) 97 , Springer, Berlin, 1979. · Zbl 0403.46022
[44] N. Linial, A. Magen, and A. Naor, Girth and Euclidean distortion , Geom. Funct. Anal. 12 (2002), 380-394. · Zbl 0991.05037 · doi:10.1007/s00039-002-8251-y
[45] M. Mendel and A. Naor, Euclidean quotients of finite metric spaces , Adv. Math. 189 (2004), 451-494. · Zbl 1088.46007 · doi:10.1016/j.aim.2003.12.001
[46] V. D. Milman and G. Schechtman, Asymptotic Theory of Finite-dimensional Normed Spaces , with an appendix by M. Gromov, Lecture Notes in Math. 1200 Springer, Berlin, 1986. · Zbl 0606.46013
[47] A. Naor and Y. Peres, Embeddings of discrete groups and the speed of random walks , Int. Math. Res. Not. IMRN 2008 , Art. ID rnn 076. · Zbl 1163.46007 · doi:10.1093/imrn/rnn076
[48] A. Naor, Y. Peres, O. Schramm, and S. Sheffield, Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces , Duke Math. J. 134 (2006), 165-197. · Zbl 1108.46012 · doi:10.1215/S0012-7094-06-13415-4
[49] A. Naor and T. Tao, Random martingales and localization of maximal inequalities , J. Funct. Anal. 259 (2010), 731-779. · Zbl 1196.42018 · doi:10.1016/j.jfa.2009.12.009
[50] K. Okikiolu, Characterization of subsets of rectifiable curves in \({\mathbf R}^ n\) , J. Lond. Math. Soc. (2) 46 (1992), 336-348. · Zbl 0758.57020 · doi:10.1112/jlms/s2-46.2.336
[51] R. Paley and A. Zygmund, A note on analytic functions in the unit circle , Math. Proc. Cambridge Philos. Soc. 28 (1932), 266-272. Zentralblatt 0005.06602 · Zbl 0005.06602
[52] A. Pełczyński, Projections in certain Banach spaces , Studia Math. 19 (1960), 209-228. · Zbl 0104.08503
[53] R. Schul, “Analyst’s traveling salesman theorems: A survey” in In the Tradition of Ahlfors-Bers, IV (Ann Arbor, Mich., 2005) , Contemp. Math. 432 , Amer. Math. Soc., Providence, 2007, 209-220. · Zbl 1187.49039
[54] Y. Stalder and A. Valette, Wreath products with the integers, proper actions and Hilbert space compression , Geom. Dedicata 124 (2007), 199-211. · Zbl 1178.20039 · doi:10.1007/s10711-006-9119-3
[55] J. M. Steele, Probability Theory and Combinatorial Optimization , CBMS-NSF Regional Conf. Ser. in Appl. Math. 69 , SIAM, Philadelphia, 1997. · Zbl 0916.90233
[56] M. Stoll, On the asymptotics of the growth of \(2\)-step nilpotent groups , J. Lond. Math. Soc. (2) 58 (1998), 38-48. · Zbl 0922.20038 · doi:10.1112/S0024610798006371
[57] R. Tessera, Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces , · Zbl 1274.43009
[58] S. Wagon, The Banach-Tarski Paradox , Cambridge Univ. Press, Cambridge, 1993. · Zbl 0569.43001
[59] J. H. Wells and L. R. Williams, Embeddings and Extensions in Analysis , Ergeb. Math. Grenzgeb. (3) 84 , Springer, New York, 1975. · Zbl 0324.46034
[60] P. Wojtaszczyk, Banach Spaces for Analysts , Cambridge Stud. Adv. Math. 25 , Cambridge Univ. Press, Cambridge, 1991. · Zbl 0724.46012
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