Startseite K-homological finiteness and hyperbolic groups
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

K-homological finiteness and hyperbolic groups

  • Heath Emerson EMAIL logo und Bogdan Nica
Veröffentlicht/Copyright: 7. April 2016
Veröffentlichen auch Sie bei De Gruyter Brill
Informationen für Autor*innen Erkunden Sie dieses Fachgebiet
Journal für die reine und angewandte Mathematik (Crelles Journal)
Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2018 Heft 745

Abstract

Motivated by classical facts concerning closed manifolds, we introduce a strong finiteness property in K-homology. We say that a C*-algebra has uniformly summable K-homology if all its K-homology classes can be represented by Fredholm modules which are finitely summable over the same dense subalgebra, and with the same degree of summability. We show that two types of C*-algebras associated to hyperbolic groups – the C*-crossed product for the boundary action, and the reduced group C*-algebra – have uniformly summable K-homology. We provide explicit summability degrees, as well as explicit finitely summable representatives for the K-homology classes.

Funding statement: The first author acknowledges support from an NSERC Discovery grant. The second author thanks the Pacific Institute for Mathematical Sciences and the Alexander von Humboldt Foundation for their support.

Acknowledgements

We thank Nigel Higson, Vadim Kaimanovich, Misha Kapovich, Bruce Kleiner, Georges Skandalis, and Bob Yuncken for correspondence or discussions. We also thank the last referee for a careful reading of the paper, and for constructive comments.

References

[1] S. Adams, Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups, Topology 33 (1994), no. 4, 765–783. 10.1016/0040-9383(94)90007-8Suche in Google Scholar

[2] C. Anantharaman-Delaroche, Purely infinite C*-algebras arising from dynamical systems, Bull. Soc. Math. France 125 (1997), no. 2, 199–225. 10.24033/bsmf.2304Suche in Google Scholar

[3] C. Anantharaman-Delaroche, Amenability and exactness for dynamical systems and their C∗-algebras, Trans. Amer. Math. Soc. 354 (2002), no. 10, 4153–4178. 10.1090/S0002-9947-02-02978-1Suche in Google Scholar

[4] U. Bader, A. Furman, T. Gelander and N. Monod, Property (T) and rigidity for actions on Banach spaces, Acta Math. 198 (2007), no. 1, 57–105. 10.1007/s11511-007-0013-0Suche in Google Scholar

[5] M. Bonk and B. Kleiner, Quasi-hyperbolic planes in hyperbolic groups, Proc. Amer. Math. Soc. 133 (2005), no. 9, 2491–2494. 10.1090/S0002-9939-05-07564-7Suche in Google Scholar

[6] A. Connes, Compact metric spaces, Fredholm modules, and hyperfiniteness, Ergodic Theory Dynam. Systems 9 (1989), no. 2, 207–220. 10.1017/S0143385700004934Suche in Google Scholar

[7] A. Connes, Noncommutative geometry, Academic Press, San Diego 1994. Suche in Google Scholar

[8] M. Coornaert, Mesures de Patterson–Sullivan sur le bord d’un espace hyperbolique au sens de Gromov, Pacific J. Math. 159 (1993), no. 2, 241–270. 10.2140/pjm.1993.159.241Suche in Google Scholar

[9] J. Cuntz, K-theory for certain C*-algebras, Ann. of Math. (2) 113 (1981), no. 1, 181–197. 10.2307/1971137Suche in Google Scholar

[10] H. Emerson, Noncommutative Poincaré duality for boundary actions of hyperbolic groups, J. reine angew. Math. 564 (2003), 1–33. 10.1515/crll.2003.090Suche in Google Scholar

[11] H. Emerson and R. Meyer, Euler characteristics and Gysin sequences for group actions on boundaries, Math. Ann. 334 (2006), no. 4, 853–904. 10.1007/s00208-005-0747-ySuche in Google Scholar

[12] H. Emerson and R. Meyer, A descent principle for the Dirac-dual-Dirac method, Topology 46 (2007), 185–209. 10.1016/j.top.2007.02.001Suche in Google Scholar

[13] H. Emerson and R. Meyer, Equivariant representable K-theory, J. Topology 46 (2009), 123–156. 10.1112/jtopol/jtp003Suche in Google Scholar

[14] H. Emerson and B. Nica, Finitely summable Fredholm modules for boundary actions of hyperbolic groups, preprint (2012), http://arxiv.org/abs/1208.0856v1. Suche in Google Scholar

[15] É. Ghys and P. de la Harpe, Sur les groupes hyperboliques d’après Mikhael Gromov, Progr. Math. 83, Birkhäuser, Basel 1990. 10.1007/978-1-4684-9167-8Suche in Google Scholar

[16] M. Goffeng and B. Mesland, Spectral triples and finite summability on Cuntz–Krieger algebras, Doc. Math. 20 (2015), 89–170. 10.4171/dm/487Suche in Google Scholar

[17] M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ. 8, Springer, New York (1987), 75–263. 10.1007/978-1-4613-9586-7_3Suche in Google Scholar

[18] N. Higson and G. Kasparov, E-theory and KK-theory for groups which act properly and isometrically on Hilbert space, Invent. Math. 144 (2001), no. 1, 23–74. 10.1007/s002220000118Suche in Google Scholar

[19] N. Higson and J. Roe, Analytic K-homology, Oxford Math. Monogr., Oxford University Press, Oxford 2000. Suche in Google Scholar

[20] P. Julg and G. Kasparov, Operator K-theory for the group SU(n, 1), J. reine angew. Math. 463 (1995), 99–152. 10.1515/crll.1995.463.99Suche in Google Scholar

[21] P. Julg and A. Valette, K-theoretic amenability for SL2(ℚp), and the action on the associated tree, J. Funct. Anal. 58 (1984), no. 2, 194–215. 10.1016/0022-1236(84)90039-9Suche in Google Scholar

[22] V. Kaimanovich, Double ergodicity of the Poisson boundary and applications to bounded cohomology, Geom. Funct. Anal. 13 (2003), no. 4, 852–861. 10.1007/s00039-003-0433-8Suche in Google Scholar

[23] I. Kapovich and N. Benakli, Boundaries of hyperbolic groups, Combinatorial and geometric group theory (New York 2000, Hoboken 2001), Contemp. Math. 296, American Mathematical Society, Providence (2002), 39–93. 10.1090/conm/296/05068Suche in Google Scholar

[24] M. Kapovich and B. Kleiner, Hyperbolic groups with low-dimensional boundary, Ann. Sci. Éc. Norm. Supér. (4) 33 (2000), no. 5, 647–669. 10.1016/S0012-9593(00)01049-1Suche in Google Scholar

[25] G. Kasparov, Lorentz groups: K-theory of unitary representations and crossed products, Dokl. Akad. Nauk SSSR 275 (1984), 541–545. Suche in Google Scholar

[26] G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91 (1988), 147–201. 10.1007/BF01404917Suche in Google Scholar

[27] G. Kasparov and G. Skandalis, Groups acting properly on “bolic” spaces and the Novikov conjecture, Ann. of Math. (2) 158 (2003), no. 1, 165–206. 10.4007/annals.2003.158.165Suche in Google Scholar

[28] B. Kleiner, The asymptotic geometry of negatively curved spaces: Uniformization, geometrization and rigidity, International congress of mathematicians. Vol. II: Invited lectures (Madrid 2006), European Mathematical Society, Zürich (2006), 743–768. 10.4171/022-2/36Suche in Google Scholar

[29] V. Lafforgue, K-théorie bivariante pour les algèbres de Banach et conjecture de Baum–Connes, Invent. Math. 149 (2002), no. 1, 1–95. 10.1007/s002220200213Suche in Google Scholar

[30] M. Laca and J. Spielberg, Purely infinite C*-algebras from boundary actions of discrete groups, J. reine angew. Math. 480 (1996), 125–139. 10.1515/crll.1996.480.125Suche in Google Scholar

[31] J. Lott, Limit sets as examples in noncommutative geometry, K-Theory 34 (2005), no. 4, 283–326. 10.1007/s10977-005-3101-ySuche in Google Scholar

[32] D. Meintrup and T. Schick, A model for the universal space for proper actions of a hyperbolic group, New York J. Math. 8 (2002), 1–7. Suche in Google Scholar

[33] R. Meyer and R. Nest, The Baum–Connes conjecture via localisation of categories, Topology 45 (2006), no. 2, 209–259. 10.1016/j.top.2005.07.001Suche in Google Scholar

[34] I. Mineyev and G. Yu, The Baum-Connes conjecture for hyperbolic groups, Invent. Math. 149 (2002), no. 1, 97–122. 10.1007/s002220200214Suche in Google Scholar

[35] B. Nica, Proper isometric actions of hyperbolic groups on Lp-spaces, Compos. Math. 149 (2013), no. 5, 773–792. 10.1112/S0010437X12000693Suche in Google Scholar

[36] M. Puschnigg, Finitely summable Fredholm modules over higher rank groups and lattices, J. K-Theory 8 (2011), no. 2, 223–239. 10.1017/is010011023jkt131Suche in Google Scholar

[37] S. Rave, On finitely summable K-homology, Dissertation, University of Münster, Münster, 2012. Suche in Google Scholar

[38] B. Simon, Trace ideals and their applications , 2nd ed., Math. Surveys Monogr. 120, American Mathematical Society, Providence 2005. Suche in Google Scholar

[39] G. Skandalis, Une notion de nucléarité en K-théorie (d’après J. Cuntz), K-Theory 1 (1988), no. 6, 549–573. 10.1007/BF00533786Suche in Google Scholar

[40] J. Väisälä, Gromov hyperbolic spaces, Expo. Math. 23 (2005), no. 3, 187–231. 10.1016/j.exmath.2005.01.010Suche in Google Scholar

[41] D. T. Wise, Cubulating small cancellation groups, Geom. Funct. Anal. 14 (2004), no. 1, 150–214. 10.1007/s00039-004-0454-ySuche in Google Scholar

[42] G. Yu, Hyperbolic groups admit proper affine isometric actions on lp-spaces, Geom. Funct. Anal. 15 (2005), no. 5, 1144–1151. 10.1007/s00039-005-0533-8Suche in Google Scholar

Received: 2015-07-04
Revised: 2015-10-28
Published Online: 2016-04-07
Published in Print: 2018-12-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Equivariant basic cohomology of Riemannian foliations
  3. Motivic decomposition of compactifications of certain group varieties
  4. A soft Oka principle for proper holomorphic embeddings of open Riemann surfaces into (ℂ*)2
  5. Associated forms and hypersurface singularities: The binary case
  6. Stratified-algebraic vector bundles
  7. Best possible rates of distribution of dense lattice orbits in homogeneous spaces
  8. K-homological finiteness and hyperbolic groups
  9. Involutions of varieties and Rost’s degree formula
  10. Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay
  11. Addendum to “Singular equivariant asymptotics and Weyl’s law”
https://doi.org/10.1515/crelle-2015-0115

Artikel in diesem Heft

  1. Frontmatter
  2. Equivariant basic cohomology of Riemannian foliations
  3. Motivic decomposition of compactifications of certain group varieties
  4. A soft Oka principle for proper holomorphic embeddings of open Riemann surfaces into (ℂ*)2
  5. Associated forms and hypersurface singularities: The binary case
  6. Stratified-algebraic vector bundles
  7. Best possible rates of distribution of dense lattice orbits in homogeneous spaces
  8. K-homological finiteness and hyperbolic groups
  9. Involutions of varieties and Rost’s degree formula
  10. Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay
  11. Addendum to “Singular equivariant asymptotics and Weyl’s law”
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 30.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/crelle-2015-0115/html?lang=de
Button zum nach oben scrollen