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Introduction to the rapid decay property. (English) Zbl 1404.20032

Brumley, Farrell (ed.) et al., Around Langlands correspondences. International conference, Université Paris Sud, Orsay, France, June 17–20, 2015. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-3573-8/pbk; 978-1-4704-4117-3/ebook). Contemp. Math. 691, 53-72 (2017).
Summary: This is an introduction to the Rapid Decay property, with a survey of known results and equivalent definitions of this property. We also discuss in details the easy case when \(G=\mathbb{Z}\). Everything in this paper is well-known by different sets of people.
For the entire collection see [Zbl 1369.11002].

MSC:

20F65 Geometric group theory
22D40 Ergodic theory on groups
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations

References:

[1] Antonescu, Cristina; Christensen, Erik, Metrics on group \(C^*\)-algebras and a non-commutative Arzel\`“a-Ascoli theorem, J. Funct. Anal., 214, 2, 247-259 (2004) · Zbl 1063.46059 · doi:10.1016/j.jfa.2004.04.015
[2] G. Arzhantseva, C. Drutu. Geometry of infinitely presented small cancellation groups, Rapid Decay and quasi-homomorphisms. arXiv:1212.5280 · Zbl 1453.20044
[3] Barr{\'e}, Sylvain; Pichot, Mika{\`“e}l, The 4-string braid group \(B_4\) has property RD and exponential mesoscopic rank, Bull. Soc. Math. France, 139, 4, 479-502 (2011) · Zbl 1266.20046
[4] Barr{\'e}, Sylvain; Pichot, Mika{\`“e}l, La propri\'”et\'e de d\'ecroissance rapide pour le groupe de Wise, Ann. Inst. Fourier (Grenoble), 65, 2, 709-724 (2015) · Zbl 1372.20039
[5] Behrstock, Jason A.; Minsky, Yair N., Centroids and the rapid decay property in mapping class groups, J. Lond. Math. Soc. (2), 84, 3, 765-784 (2011) · Zbl 1269.20032 · doi:10.1112/jlms/jdr027
[6] Bhatia, Rajendra, Fourier series, Texts and Readings in Mathematics, vi+106 pp. (1993), Hindustan Book Agency, New Delhi · Zbl 0940.42001
[7] N. Bourbaki, Espaces vectoriels topologiques. Chapitres 1 a 5., Elements de mathematiques. · Zbl 1106.46003
[8] Brodzki, Jacek; Niblo, Graham A., Approximation properties for discrete groups. \(C^\ast \)-algebras and elliptic theory, Trends Math., 23-35 (2006), Birkh\`“auser, Basel · Zbl 1119.46054 · doi:10.1007/978-3-7643-7687-1\_2
[9] Bowditch, Brian H., Embedding median algebras in products of trees, Geom. Dedicata, 170, 157-176 (2014) · Zbl 1330.20062 · doi:10.1007/s10711-013-9874-x
[10] Boyer, Adrien, Quasi-regular representations and rapid decay, Potential Anal., 44, 2, 355-372 (2016) · Zbl 1333.22008 · doi:10.1007/s11118-015-9515-0
[11] Boyer, Adrien, Semisimple Lie groups satisfy property RD, a short proof, C. R. Math. Acad. Sci. Paris, 351, 9-10, 335-338 (2013) · Zbl 1312.22008 · doi:10.1016/j.crma.2013.05.007
[12] Candellero, Elisabetta; Gilch, Lorenz A., Phase transitions for random walk asymptotics on free products of groups, Random Structures Algorithms, 40, 2, 150-181 (2012) · Zbl 1242.05251 · doi:10.1002/rsa.20370
[13] Cartwright, Donald I., Some examples of random walks on free products of discrete groups, Ann. Mat. Pura Appl. (4), 151, 1-15 (1988) · Zbl 0661.60018 · doi:10.1007/BF01762785
[14] Chatterji, Indira, Property (RD) for cocompact lattices in a finite product of rank one Lie groups with some rank two Lie groups, Geom. Dedicata, 96, 161-177 (2003) · Zbl 1012.22018 · doi:10.1023/A:1022184513930
[15] Chatterji, I.; Ruane, K., Some geometric groups with rapid decay, Geom. Funct. Anal., 15, 2, 311-339 (2005) · Zbl 1134.22005 · doi:10.1007/s00039-005-0508-9
[16] Chatterji, I.; Pittet, C.; Saloff-Coste, L., Connected Lie groups and property RD, Duke Math. J., 137, 3, 511-536 (2007) · Zbl 1119.22006 · doi:10.1215/S0012-7094-07-13733-5
[17] I. Chatterji and L. Saloff-Coste. Introduction to the property of Rapid decay. Notes not intended for publication, available at: http://www.aimath.org/WWN/rapiddecay/IntroRD.pdf
[18] I. Chatterji, Ch. Pittet and L. Saloff-Coste. On the decay of the heat kernel. Work in progress. · Zbl 0985.60043
[19] Ciobanu, Laura; Holt, Derek F.; Rees, Sarah, Rapid decay is preserved by graph products, J. Topol. Anal., 5, 2, 225-237 (2013) · Zbl 1286.43002 · doi:10.1142/S1793525313500052
[20] Ciobanu, Laura; Holt, Derek F.; Rees, Sarah, Rapid decay and Baum-Connes for large type Artin groups, Trans. Amer. Math. Soc., 368, 9, 6103-6129 (2016) · Zbl 1383.20020 · doi:10.1090/tran/6532
[21] Dru{\c{t}}u, Cornelia; Sapir, Mark, Relatively hyperbolic groups with rapid decay property, Int. Math. Res. Not., 19, 1181-1194 (2005) · Zbl 1077.22006 · doi:10.1155/IMRN.2005.1181
[22] Connes, Alain, Noncommutative geometry, xiv+661 pp. (1994), Academic Press, Inc., San Diego, CA · Zbl 0818.46076
[23] Connes, Alain; Moscovici, Henri, Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology, 29, 3, 345-388 (1990) · Zbl 0759.58047 · doi:10.1016/0040-9383(90)90003-3
[24] Garncarek, {\L }ukasz, Property of rapid decay for extensions of compactly generated groups, Publ. Mat., 59, 2, 301-312 (2015) · Zbl 1321.22009
[25] L. Garncarek. Mini-course: property of rapid decay. https://arxiv.org/abs/1603.06730 · Zbl 1321.22009
[26] F. Gautero. Rapid Decay and 3-manifold groups. Preprint 2016. · Zbl 1098.20032
[27] Gou{\`“e}zel, S{\'”e}bastien, Local limit theorem for symmetric random walks in Gromov-hyperbolic groups, J. Amer. Math. Soc., 27, 3, 893-928 (2014) · Zbl 1320.60017 · doi:10.1090/S0894-0347-2014-00788-8
[28] Grigorchuk, Rostislav; Nagnibeda, Tatiana, Complete growth functions of hyperbolic groups, Invent. Math., 130, 1, 159-188 (1997) · Zbl 0880.20024 · doi:10.1007/s002220050181
[29] Haagerup, Uffe, An example of a nonnuclear \(C^{\ast } \)-algebra, which has the metric approximation property, Invent. Math., 50, 3, 279-293 (1978/79) · Zbl 0408.46046 · doi:10.1007/BF01410082
[30] de la Harpe, Pierre, Groupes hyperboliques, alg\`“ebres d”op\'erateurs et un th\'eor\`“eme de Jolissaint, C. R. Acad. Sci. Paris S\'”er. I Math., 307, 14, 771-774 (1988) · Zbl 0653.46059
[31] de la Harpe, Pierre; Robertson, A. Guyan; Valette, Alain, On the spectrum of the sum of generators of a finitely generated group. II, Colloq. Math., 65, 1, 87-102 (1993) · Zbl 0846.46036
[32] Herz, Carl, Sur le ph\'enom\`“ene de Kunze-Stein, C. R. Acad. Sci. Paris S\'”er. A-B, 271, A491-A493 (1970) · Zbl 0198.18202
[33] Ji, Ronghui; Schweitzer, Larry B., Spectral invariance of smooth crossed products, and rapid decay locally compact groups, \(K\)-Theory, 10, 3, 283-305 (1996) · Zbl 0854.46052 · doi:10.1007/BF00538186
[34] Jolissaint, Paul, Rapidly decreasing functions in reduced \(C^*\)-algebras of groups, Trans. Amer. Math. Soc., 317, 1, 167-196 (1990) · Zbl 0711.46054 · doi:10.2307/2001458
[35] Jolissaint, Paul, \(K\)-theory of reduced \(C^*\)-algebras and rapidly decreasing functions on groups, \(K\)-Theory, 2, 6, 723-735 (1989) · Zbl 0692.46062 · doi:10.1007/BF00538429
[36] Lafforgue, Vincent, A proof of property (RD) for cocompact lattices of \({\rm SL}(3,\mathbf{R})\) and \({\rm SL}(3,\mathbf{C})\), J. Lie Theory, 10, 2, 255-267 (2000) · Zbl 0981.46046
[37] Lafforgue, Vincent, \(K\)-th\'eorie bivariante pour les alg\`“ebres de Banach et conjecture de Baum-Connes, Invent. Math., 149, 1, 1-95 (2002) · Zbl 1084.19003 · doi:10.1007/s002220200213
[38] Lalley, Steven P., Finite range random walk on free groups and homogeneous trees, Ann. Probab., 21, 4, 2087-2130 (1993) · Zbl 0804.60006
[39] Leptin, H., On locally compact groups with invariant means, Proc. Amer. Math. Soc., 19, 489-494 (1968) · Zbl 0155.05702
[40] Lubotzky, Alexander; Mozes, Shahar; Raghunathan, M. S., Cyclic subgroups of exponential growth and metrics on discrete groups, C. R. Acad. Sci. Paris S\'er. I Math., 317, 8, 735-740 (1993) · Zbl 0786.22016
[41] Pak, Igor; Smirnova-Nagnibeda, Tatiana, On non-uniqueness of percolation on nonamenable Cayley graphs, C. R. Acad. Sci. Paris S\'er. I Math., 330, 6, 495-500 (2000) · Zbl 0947.43003 · doi:10.1016/S0764-4442(00)00211-1
[42] Nevo, Amos, Spectral transfer and pointwise ergodic theorems for semi-simple Kazhdan groups, Math. Res. Lett., 5, 3, 305-325 (1998) · Zbl 0942.22007 · doi:10.4310/MRL.1998.v5.n3.a5
[43] Nevo, Amos, On discrete groups and pointwise ergodic theory. Random walks and discrete potential theory, Cortona, 1997, Sympos. Math., XXXIX, 279-305 (1999), Cambridge Univ. Press, Cambridge · Zbl 0976.22006
[44] Nica, Bogdan, On the degree of rapid decay, Proc. Amer. Math. Soc., 138, 7, 2341-2347 (2010) · Zbl 1247.22010 · doi:10.1090/S0002-9939-10-10289-5
[45] B. Nica. On operator norms for hyperbolic groups. To appear in Journal of Topology and Analysis. · Zbl 1365.20041
[46] D. V. Osin. Property (RD) for C’(1/6)-groups. Preprint 2014.
[47] M. Perrone. Rapid Decay and weak containment of unitary representations. Unpublished.
[48] Perrone, Mattia, Radial rapid decay property for cocompact lattices, J. Funct. Anal., 256, 11, 3471-3489 (2009) · Zbl 1168.22006 · doi:10.1016/j.jfa.2009.02.007
[49] Rajagopalan, M., On the \(L^p\)-space of a locally compact group, Colloq. Math., 10, 49-52 (1963) · Zbl 0117.34102
[50] Ramagge, J.; Robertson, G.; Steger, T., A Haagerup inequality for \(\widetilde A_1\times \widetilde A_1\) and \(\widetilde A_2\) buildings, Geom. Funct. Anal., 8, 4, 702-731 (1998) · Zbl 0906.43009 · doi:10.1007/s000390050071
[51] Sapir, Mark, The rapid decay property and centroids in groups, J. Topol. Anal., 7, 3, 513-541 (2015) · Zbl 1334.20038 · doi:10.1142/S1793525315500247
[52] Talbi, Malik, A Haagerup inequality, deformation of triangles and affine buildings, J. Inst. Math. Jussieu, 5, 2, 187-227 (2006) · Zbl 1172.43300 · doi:10.1017/S1474748005000241
[53] Valette, Alain, Introduction to the Baum-Connes conjecture, Lectures in Mathematics ETH Z\`“urich, x+104 pp. (2002), Birkh\'”auser Verlag, Basel · Zbl 1136.58013 · doi:10.1007/978-3-0348-8187-6
[54] Valette, Alain, On the Haagerup inequality and groups acting on \(\widetilde A_n\)-buildings, Ann. Inst. Fourier (Grenoble), 47, 4, 1195-1208 (1997) · Zbl 0886.51003
[55] Woess, Wolfgang, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics 138, xii+334 pp. (2000), Cambridge University Press, Cambridge · Zbl 1142.60003 · doi:10.1017/CBO9780511470967
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