K-homological finiteness and hyperbolic groups

Heath Emerson; Bogdan Nica

doi:10.1515/crelle-2015-0115

Article

K-homological finiteness and hyperbolic groups

Heath Emerson and Bogdan Nica

Published/Copyright: April 7, 2016

Published by

Become an author with De Gruyter Brill

Author Information Explore this Subject

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2018 Issue 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)90007-8Search 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.2304Search 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/S0002-9947-02-02978-1Search 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/s11511-007-0013-0Search 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/S0002-9939-05-07564-7Search 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/S0143385700004934Search in Google Scholar

[7] A. Connes, Noncommutative geometry, Academic Press, San Diego 1994. Search 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.1993.159.241Search in Google Scholar

[9] J. Cuntz, K-theory for certain C*-algebras, Ann. of Math. (2) 113 (1981), no. 1, 181–197. 10.2307/1971137Search 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.2003.090Search 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-ySearch 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.2007.02.001Search in Google Scholar

[13] H. Emerson and R. Meyer, Equivariant representable K-theory, J. Topology 46 (2009), 123–156. 10.1112/jtopol/jtp003Search 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. Search 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. 10.1007/978-1-4684-9167-8Search 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/487Search 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_3Search 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/s002220000118Search in Google Scholar

[19] N. Higson and J. Roe, Analytic K-homology, Oxford Math. Monogr., Oxford University Press, Oxford 2000. Search 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.1995.463.99Search 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)90039-9Search 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/s00039-003-0433-8Search 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/296/05068Search 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)01049-1Search in Google Scholar

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

[26] G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91 (1988), 147–201. 10.1007/BF01404917Search 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.2003.158.165Search 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. 10.4171/022-2/36Search 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/s002220200213Search 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.1996.480.125Search 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-ySearch 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. Search 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.2005.07.001Search 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/s002220200214Search 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/S0010437X12000693Search 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/is010011023jkt131Search in Google Scholar

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

[38] B. Simon, Trace ideals and their applications , 2nd ed., Math. Surveys Monogr. 120, American Mathematical Society, Providence 2005. Search 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/BF00533786Search in Google Scholar

[40] J. Väisälä, Gromov hyperbolic spaces, Expo. Math. 23 (2005), no. 3, 187–231. 10.1016/j.exmath.2005.01.010Search 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-ySearch 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/s00039-005-0533-8Search in Google Scholar

Received: 2015-07-04

Revised: 2015-10-28

Published Online: 2016-04-07

Published in Print: 2018-12-01

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/crelle-2015-0115