Abstract
We introduce classifying spaces for families of subgroups and deduce their homotopy theoretic characterization. Afterwards, we extend the notion of von Neumann dimension to arbitrary modules over the group von Neumann algebra. This leads to the definition of ℓ 2-Betti numbers of arbitrary G-spaces and hence allows the definition of ℓ 2-Betti numbers of general discrete countable groups via classifying spaces. We give example computations for various groups and explain how ℓ 2-Betti numbers detect finitely co-Hopfian groups and how they yield a bound on the deficiency of finitely presented groups.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
W. Dicks, P.A. Linnell, L 2-Betti numbers of one-relator groups. Math. Ann. 337(4), 855–874 (2007). MR 2285740
D. Gaboriau, Invariants l 2 de relations d’équivalence et de groupes. Publ. Math. Inst. Hautes Études Sci. 95, 93–150 (2002) (French). MR 1953191
M. Gromov, Asymptotic invariants of infinite groups, in Geometric Group Theory, vol. 2 (Sussex, 1991). London Mathematical Society. Lecture Notes Series, vol. 182 (Cambridge University Press, Cambridge, 1993), pp. 1–295. MR 1253544
F. Hirzebruch, Automorphe Formen und der Satz von Riemann-Roch, in Symposium Internacional de Topología Algebraica International Symposium on Algebraic Topology (Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958), pp. 129–144 (German). MR 0103280
D. Kyed, H.D. Petersen, S. Vaes, L 2-Betti numbers of locally compact groups and their cross section equivalence relations. Trans. Am. Math. Soc. 367(7), 4917–4956 (2015). MR 3335405
J. Lott, Deficiencies of lattice subgroups of Lie groups. Bull. Lond. Math. Soc. 31(2), 191–195 (1999). MR 1664196
W. Lück, Dimension theory of arbitrary modules over finite von Neumann algebras and L 2-Betti numbers. I. Foundations. J. Reine Angew. Math. 495, 135–162 (1998). MR 1603853
W. Lück, L 2-Invariants: Theory and Applications to Geometry and K-Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 44 (Springer, Berlin, 2002). MR 1926649
W. Lück, D. Osin, Approximating the first L 2-Betti number of residually finite groups. J. Topol. Anal. 3(2), 153–160 (2011). MR 2819192
R.C. Lyndon, P.E. Schupp, Combinatorial Group Theory. Classics in Mathematics (Springer, Berlin, 2001). Reprint of the 1977 edition. MR 1812024
H.D. Petersen, L2-Betti numbers of locally compact groups, PhD Thesis, Department of Mathematical Sciences, Faculty of Science, University of Copenhagen, 2012. http://www.math.ku.dk/noter/filer/phd13hdp.pdf
H.D. Petersen, A. Valette, L 2-Betti numbers and Plancherel measure. J. Funct. Anal. 266(5), 3156–3169 (2014). MR 3158720
M. Pichot, Semi-continuity of the first l 2-Betti number on the space of finitely generated groups. Comment. Math. Helv. 81(3), 643–652 (2006). MR 2250857
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Kammeyer, H. (2019). ℓ 2-Betti Numbers of Groups. In: Introduction to ℓ²-invariants. Lecture Notes in Mathematics, vol 2247. Springer, Cham. https://doi.org/10.1007/978-3-030-28297-4_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-28297-4_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-28296-7
Online ISBN: 978-3-030-28297-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)