The Baum-Connes conjecture with coefficients for word-hyperbolic groups

Puschnigg, Michael

Séminaire Bourbaki volume 2012/2013 : exposés 1059-1073 - Avec table par noms d'auteurs de 1948/49 à 2012/13, Astérisque, no. 361 (2014), Exposé no. 1062, 34 p.

MR Zbl

@incollection{AST_2014__361__115_0,
     author = {Puschnigg, Michael},
     title = {The {Baum-Connes} conjecture with coefficients for word-hyperbolic groups},
     booktitle = {S\'eminaire Bourbaki volume 2012/2013 : expos\'es 1059-1073 - Avec table par noms d'auteurs de 1948/49 \`a 2012/13},
     series = {Ast\'erisque},
     note = {talk:1062},
     pages = {115--148},
     year = {2014},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {361},
     mrnumber = {3289279},
     zbl = {1357.19005},
     language = {en},
     url = {https://www.numdam.org/item/AST_2014__361__115_0/}
}

TY  - CHAP
AU  - Puschnigg, Michael
TI  - The Baum-Connes conjecture with coefficients for word-hyperbolic groups
BT  - Séminaire Bourbaki volume 2012/2013 : exposés 1059-1073 - Avec table par noms d'auteurs de 1948/49 à 2012/13
AU  - Collectif
T3  - Astérisque
N1  - talk:1062
PY  - 2014
SP  - 115
EP  - 148
IS  - 361
PB  - Société mathématique de France
UR  - https://www.numdam.org/item/AST_2014__361__115_0/
LA  - en
ID  - AST_2014__361__115_0
ER  -

%0 Book Section
%A Puschnigg, Michael
%T The Baum-Connes conjecture with coefficients for word-hyperbolic groups
%B Séminaire Bourbaki volume 2012/2013 : exposés 1059-1073 - Avec table par noms d'auteurs de 1948/49 à 2012/13
%A Collectif
%S Astérisque
%Z talk:1062
%D 2014
%P 115-148
%N 361
%I Société mathématique de France
%U https://www.numdam.org/item/AST_2014__361__115_0/
%G en
%F AST_2014__361__115_0

Puschnigg, Michael. The Baum-Connes conjecture with coefficients for word-hyperbolic groups, dans Séminaire Bourbaki volume 2012/2013 : exposés 1059-1073 - Avec table par noms d'auteurs de 1948/49 à 2012/13, Astérisque, no. 361 (2014), Exposé no. 1062, 34 p.. https://www.numdam.org/item/AST_2014__361__115_0/

Bibliographie
Cité par

[1] G. Arzhantseva & T. Delzant - "Examples of random groups", preprint, 2008.

[2] M. F. Atiyah - "Bott periodicity and the index of elliptic operators", Quart. J. Math. Oxford Ser. (2) 19 (1968), p. 113-140. | MR | Zbl | DOI

[3] M. F. Atiyah & I. M. Singer - "The index of elliptic operators I", Ann. of Math. (2) 87 (1968), p. 484-530. | MR | Zbl | DOI

[4] M. F. Atiyah & I. M. Singer, "The index of elliptic operators IV", Ann. of Math. (2) 93 (1971), p. 119-138. | MR | Zbl | DOI

[5] S. Baaj & P. Julg - "Théorie bivariante de Kasparov et opérateurs non bornés dans les $C^{*}$ -modules hilbertiens", C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 21, p. 875-878. | MR | Zbl

[6] U. Bader, A. Furman, T. Gelander & N. Monod - "Property $(T)$ and rigidity for actions on Banach spaces", Acta Math. 198 (2007), no. 1, p. 57-105. | MR | Zbl | DOI

[7] P. Baum & A. Connes - "Geometric $K$ -theory for Lie groups and foliations", Enseign. Math. (2) 46 (2000), no. 1-2, p. 3-42. | MR | Zbl

[8] P. Baum, A. Connes & N. Higson - "Classifying space for proper actions and $K$ -theory of group $C^{*}$ -algebras", in $C^{*}$ -algebras: 1943-1993 (San Antonio, 1993), Contemp. Math., vol. 167, Amer. Math. Soc., Providence, 1994, p. 240-291. | MR | Zbl

[9] B. Bekka, P. De La Harpe & A. Valette - Kazhdan's property $(T)$ , Cambridge Univ. Press, Cambridge, 2008. | MR | Zbl | DOI

[10] J.-B. Bost - "Principe d'Oka, $K$ -théorie et systèmes dynamiques non commutatifs", Invent. Math. 101 (1990), no. 2, p. 261-333. | MR | Zbl | EuDML | DOI

[11] J. Bourgain - "A Hausdorff-Young inequality for $B$ -convex Banach spaces", Pacific J. Math., 101 (1982), no. 2, p. 255-262. | MR | Zbl | DOI

[12] J. Chabert, S. Echterhoff & R. Nest - "The Connes-Kasparov conjecture for almost connected groups and for linear $p$ -adic groups", Publ. Math. IHÉS (2003), no. 97, p. 239-278. | MR | Zbl | EuDML | Numdam | DOI

[13] A. Connes - Noncommutative geometry, Academic Press Inc., San Diego, 1994. | MR | Zbl

[14] J. Cuntz - "A new look at $K K$ -theory", $K$ -Theory 1 (1987), no. 1, p. 31-51. | MR | Zbl | DOI

[15] T. Fack - " $K$ -théorie bivariante de Kasparov", in Séminaire Bourbaki, vol. 1982/83, Astérisque, vol. 105-106, Soc. Math. France, Paris, 1983, p. 149-166. | MR | Zbl | EuDML | Numdam

[16] É. Ghys - "Groupes aléatoires (d'après Misha Gromov,...)", in Séminaire Bourbaki, vol. 2002/03, Astérisque, vol. 294, Soc. Math. France, Paris, 2004, p. 173-204. | MR | Zbl | EuDML | Numdam

[17] M. Gromov - "Hyperbolic groups", in Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, p. 75-263. | MR | Zbl | DOI

[18] M. Gromov, "Asymptotic invariants of infinite groups", in Geometric group theory (Sussex, 1991) II, London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, p. 1-295. | MR | Zbl

[19] M. Gromov, "Random walk in random groups", Geom. Fund. Anal. 13 (2003), no. 1, p. 73-146. | Zbl | MR | DOI

[20] P. De La Harpe & A. Valette - La propriété $(T)$ de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger), Astérisque, vol. 175, Soc. Math. France, Paris, 1989. | MR | Zbl | Numdam

[21] N. Higson - "A characterization of $K K$ -theory", Pacific J. Math. 126 (1987), no. 2, p. 253-276. | MR | Zbl | DOI

[22] N. Higson, "The Baum-Connes conjecture", in Proceedings of the International Congress of Mathematicians (Berlin, 1998), no. extra vol. II, 1998, p. 637-646. | MR | Zbl | EuDML

[23] N. Higson & G. Kasparov - " $E$ -theory and $K K$ -theory for groups which act properly and isometrically on Hilbert space", Invent. Math. 144 (2001), no. 1, p. 23-74. | MR | Zbl | DOI

[24] N. Higson, V. Lafforgue & G. Skandalis - "Counterexamples to the Baum-Connes conjecture", Geom. Funct. Anal. 12 (2002), no. 2, p. 330-354. | MR | Zbl | DOI

[25] R. E. Howe & C. C. Moore - "Asymptotic properties of unitary representations", J. Funct. Anal. 32 (1979), no. 1, p. 72-96. | MR | Zbl | DOI

[26] P. Julg - "Remarks on the Baum-Connes conjecture and Kazhdan's property $T$ ", in Operator algebras and their applications (Waterloo, 1994/1995), Fields Inst. Commun., vol. 13, Amer. Math. Soc., Providence, 1997, p. 145-153. | MR | Zbl

[27] P. Julg, "Travaux de N. Higson et G. Kasparov sur la conjecture de Baum-Connes", in Séminaire Bourbaki, vol. 1997/98, Astérisque, vol. 252, Soc. Math. France, Paris, 1998, p. 151-183. | MR | Zbl | EuDML | Numdam

[28] P. Julg, "La conjecture de Baum-Connes à coefficients pour le groupe $S p (n, 1)$ ", C. R. Math. Acad. Sci. Paris 334 (2002), no. 7, p. 533-538. | MR | Zbl | DOI

[29] P. Julg & A. Valette - " $K$ -theoretic amenability for $S L_{2} (𝐐_{p})$ , and the action on the associated tree", J. Funct. Anal. 58 (1984), no. 2, p. 194-215. | MR | Zbl | DOI

[30] G. Kasparov - "The operator $K$ -functor and extensions of $C^{*}$ -algebras", Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), no. 3, p. 571-636. | MR | Zbl

[31] G. Kasparov, "Operator $K$ -theory and its applications: elliptic operators, group representations, higher signatures, $C^{*}$ -extensions", in International Congress of Mathematicians Proceedings (Warsaw, 1983), PWN, Warsaw, 1984, p. 987-1000. | MR | Zbl

[32] G. Kasparov, "Equivariant $K K$ -theory and the Novikov conjecture", Invent. Math. 91 (1988), no. 1, p. 147-201. | MR | Zbl | EuDML | DOI

[33] G. Kasparov & G. Skandalis - "Groups acting properly on "bolic" spaces and the Novikov conjecture", Ann. of Math. (2) 158 (2003), no. 1, p. 165-206. | MR | Zbl | DOI

[34] G. Kasparov & G. Yu - "The Novikov conjecture and geometry of Banach spaces", Geom. Topol. 16 (2012), no. 3, p. 1859-1880. | MR | Zbl | DOI

[35] V. Lafforgue - "Banach $K K$ -theory and the Baum-Connes conjecture", in International Congress of Mathematicians Proceedings (Beijing, 2002) II, Higher Ed. Press, Beijing, 2002, p. 795-812. | MR | Zbl

[36] V. Lafforgue, " $K$ -théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes", Invent. Math. 149 (2002), no. 1, p. 1-95. | MR | Zbl | DOI

[37] V. Lafforgue, " $K$ -théorie bivariante pour les algèbres de Banach, groupoïdes et conjecture de Baum-Connes (avec un appendice d'Hervé Oyono-Oyono)", J. Inst. Math. Jussieu 6 (2007), no. 3, p. 415-451. | MR | Zbl | DOI

[38] V. Lafforgue, "Un renforcement de la propriété $(T)$ ", Duke Math. J. 143 (2008), no. 3, p. 559-602. | MR | Zbl | DOI

[39] V. Lafforgue, "Propriété $(T)$ renforcée banachique et transformation de Fourier rapide", J. Topol. Anal. 1 (2009), no. 3, p. 191-206. | MR | Zbl | DOI

[40] V. Lafforgue, "Propriété $(T)$ renforcée et conjecture de Baum-Connes", in Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence, 2010, p. 323-345. | MR | Zbl

[41] V. Lafforgue, "La conjecture de Baum-Connes à coefficients pour les groupes hyperboliques", J. Noncommut. Geom. 6 (2012), no. 1, p. 1-197. | MR | Zbl | DOI

[42] G. A. Margulis - Discrete subgroups of semisimple Lie groups, Ergeb. Math. Grenzgeb. (3), vol. 17, Springer-Verlag, Berlin, 1991. | Zbl | MR

[43] M. Mendel & A. Naor - "Towards a calculus for non-linear spectral gaps", in Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, Soc. Ind. Appl. Math., Philadelphie, 2010, p. 236-255. | Zbl | DOI

[44] M. Mendel & A. Naor, "Non-linear spectral calculus and super-expanders", arXiv:1207.4705, 2012. | MR

[45] R. Meyer - "Equivariant Kasparov theory and generalized homomorphisms", $K$ -Theory 21 (2000), no. 3, p. 201-228. | Zbl | MR | DOI

[46] I. Mineyev & G. Yu - "The Baum-Connes conjecture for hyperbolic groups", Invent. Math. 149 (2002), no. 1, p. 97-122. | Zbl | MR | DOI

[47] C. Phillips - Equivariant $K$ -theory for proper actions, Pitman Res. Notes Math., vol. 178, Longman Scientific and Technical, Essex, 1989. | Zbl | MR

[48] R. T. Powers - "Simplicity of the $C^{*}$ -algebra associated with the free group on two generators", Duke Math. J. 42 (1975), p. 151-156. | MR | Zbl | DOI

[49] G. Skandalis - "Une notion de nucléarité en $K$ -théorie (d'après J. Cuntz)", $K$ -Theory 1 (1988), no. 6, p. 549-573. | MR | Zbl | DOI

[50] G. Skandalis, "Progrès récents sur la conjecture de Baum-Connes. Contribution de Vincent Lafforgue", in Séminaire Bourbaki, vol. 1999/2000, Astérisque, vol. 276, Soc. Math. France, Paris, 2002, p. 105-135. | MR | Zbl | EuDML | Numdam

[51] K. Thomsen - "The universal property of equivariant $K K$ -theory", J. reine angew. Math. 504 (1998), p. 55-71. | Zbl | MR

[52] J.-L. Tu - "The Baum-Connes conjecture and discrete group actions on trees", K-Theory 17 (1999), no. 4, p. 303-318. | MR | Zbl | DOI

[53] A. Wassermann - "Une démonstration de la conjecture de Connes-Kasparov pour les groupes de Lie linéaires connexes réductifs", C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), no. 18, p. 559-562. | MR | Zbl

[54] G. Yu - "Hyperbolic groups admit proper affine isometric actions on $l^{p}$ -spaces", Geom. Funct. Anal. 15 (2005), no. 5, p. 1144-1151. | MR | Zbl | DOI