𝐿²-Betti numbers of locally compact groups and their cross section equivalence relations

Kyed, David; Petersen, Henrik; Vaes, Stefaan

doi:10.1090/S0002-9947-2015-06449-6

$L^2$-Betti numbers of locally compact groups and their cross section equivalence relations
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Trans. Amer. Math. Soc. 367 (2015), 4917-4956 Request permission

Abstract:

We prove that the $L^2$-Betti numbers of a unimodular locally compact group $G$ coincide, up to a natural scaling constant, with the $L^2$-Betti numbers of the countable equivalence relation induced on a cross section of any essentially free ergodic probability measure preserving action of $G$. As a consequence, we obtain that the reduced and unreduced $L^2$-Betti numbers of $G$ agree and that the $L^2$-Betti numbers of a lattice $\Gamma$ in $G$ equal those of $G$ up to scaling by the covolume of $\Gamma$ in $G$. We also deduce several vanishing results, including the vanishing of the reduced $L^2$-cohomology for amenable locally compact groups.

References

Scot Adams, George A. Elliott, and Thierry Giordano, Amenable actions of groups, Trans. Amer. Math. Soc. 344 (1994), no. 2, 803–822. MR 1250814, DOI 10.1090/S0002-9947-1994-1250814-5
C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, Monographies de L’Enseignement Mathématique [Monographs of L’Enseignement Mathématique], vol. 36, L’Enseignement Mathématique, Geneva, 2000. With a foreword by Georges Skandalis and Appendix B by E. Germain. MR 1799683
M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974) Astérisque, No. 32-33, Soc. Math. France, Paris, 1976, pp. 43–72. MR 0420729
Uri Bader, Alex Furman, and Roman Sauer, Integrable measure equivalence and rigidity of hyperbolic lattices, Invent. Math. 194 (2013), no. 2, 313–379. MR 3117525, DOI 10.1007/s00222-012-0445-9
U. Bader, A. Furman and R. Sauer, Weak notions of normality and vanishing up to rank of $L^2$-Betti numbers. Int. Math. Res. Notices (2013), doi:10.1093/imrn/rnt029.
Philippe Blanc, Sur la cohomologie continue des groupes localement compacts, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 2, 137–168 (French). MR 543215, DOI 10.24033/asens.1364
Bachir Bekka, Pierre de la Harpe, and Alain Valette, Kazhdan’s property (T), New Mathematical Monographs, vol. 11, Cambridge University Press, Cambridge, 2008. MR 2415834, DOI 10.1017/CBO9780511542749
Armand Borel and Nolan R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, No. 94, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. MR 554917
Jeff Cheeger and Mikhael Gromov, $L_2$-cohomology and group cohomology, Topology 25 (1986), no. 2, 189–215. MR 837621, DOI 10.1016/0040-9383(86)90039-X
Alain Connes, Sur la théorie non commutative de l’intégration, Algèbres d’opérateurs (Sém., Les Plans-sur-Bex, 1978) Lecture Notes in Math., vol. 725, Springer, Berlin, 1979, pp. 19–143 (French). MR 548112
A. Connes, J. Feldman, and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynam. Systems 1 (1981), no. 4, 431–450 (1982). MR 662736, DOI 10.1017/s014338570000136x
Alain Connes and Dimitri Shlyakhtenko, $L^2$-homology for von Neumann algebras, J. Reine Angew. Math. 586 (2005), 125–168. MR 2180603, DOI 10.1515/crll.2005.2005.586.125
Patrick Delorme, $1$-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations, Bull. Soc. Math. France 105 (1977), no. 3, 281–336 (French). MR 578893, DOI 10.24033/bsmf.1853
Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), no. 2, 289–324. MR 578656, DOI 10.1090/S0002-9947-1977-0578656-4
Peter Forrest, On the virtual groups defined by ergodic actions of $R^{n}$ and $\textbf {Z}^{n}$, Advances in Math. 14 (1974), 271–308. MR 417388, DOI 10.1016/0001-8708(74)90033-4
Damien Gaboriau, Coût des relations d’équivalence et des groupes, Invent. Math. 139 (2000), no. 1, 41–98 (French, with English summary). MR 1728876, DOI 10.1007/s002229900019
Damien Gaboriau, Invariants $l^2$ de relations d’équivalence et de groupes, Publ. Math. Inst. Hautes Études Sci. 95 (2002), 93–150 (French). MR 1953191, DOI 10.1007/s102400200002
A. Guichardet, Cohomologie des groupes topologiques et des algèbres de Lie, Textes Mathématiques [Mathematical Texts], vol. 2, CEDIC, Paris, 1980 (French). MR 644979
David Kyed, $L^2$-homology for compact quantum groups, Math. Scand. 103 (2008), no. 1, 111–129. MR 2464704, DOI 10.7146/math.scand.a-15072
Wolfgang Lück, Dimension theory of arbitrary modules over finite von Neumann algebras and $L^2$-Betti numbers. I. Foundations, J. Reine Angew. Math. 495 (1998), 135–162. MR 1603853, DOI 10.1515/crll.1998.015
Wolfgang Lück, $L^2$-invariants: theory and applications to geometry and $K$-theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 44, Springer-Verlag, Berlin, 2002. MR 1926649, DOI 10.1007/978-3-662-04687-6
Henrik Densing Petersen, $L^2$-Betti numbers of locally compact groups, C. R. Math. Acad. Sci. Paris 351 (2013), no. 9-10, 339–342 (English, with English and French summaries). MR 3072156, DOI 10.1016/j.crma.2013.05.013
Jesse Peterson and Andreas Thom, Group cocycles and the ring of affiliated operators, Invent. Math. 185 (2011), no. 3, 561–592. MR 2827095, DOI 10.1007/s00222-011-0310-2
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
Florian Martin, Reduced 1-cohomology of connected locally compact groups and applications, J. Lie Theory 16 (2006), no. 2, 311–328. MR 2197595
Niels Meesschaert, Sven Raum, and Stefaan Vaes, Stable orbit equivalence of Bernoulli actions of free groups and isomorphism of some of their factor actions, Expo. Math. 31 (2013), no. 3, 274–294. MR 3108102, DOI 10.1016/j.exmath.2012.08.012
G. D. Mostow, Cohomology of topological groups and solvmanifolds, Ann. of Math. (2) 73 (1961), 20–48. MR 125179, DOI 10.2307/1970281
Roman Sauer, $L^2$-Betti numbers of discrete measured groupoids, Internat. J. Algebra Comput. 15 (2005), no. 5-6, 1169–1188. MR 2197826, DOI 10.1142/S0218196705002748
Roman Sauer and Andreas Thom, A spectral sequence to compute $L^2$-Betti numbers of groups and groupoids, J. Lond. Math. Soc. (2) 81 (2010), no. 3, 747–773. MR 2650795, DOI 10.1112/jlms/jdq017
Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728, DOI 10.1007/978-1-4612-6188-9
Andreas Thom, $L^2$-cohomology for von Neumann algebras, Geom. Funct. Anal. 18 (2008), no. 1, 251–270. MR 2399103, DOI 10.1007/s00039-007-0634-7
V. S. Varadarajan, Groups of automorphisms of Borel spaces, Trans. Amer. Math. Soc. 109 (1963), 191–220. MR 159923, DOI 10.1090/S0002-9947-1963-0159923-5
Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. MR 776417, DOI 10.1007/978-1-4684-9488-4