Amenable groups, topological entropy and Betti numbers. (English) Zbl 1016.37009
Let \(G\) be a finitely generated amenable group and \(\Sigma_G =\{0,1\}^G\) the Bernoulli shiftspace over the field of two elements. The author investigates linear, \(G\)-invariant subspaces in the spirit of Atiyah’s \(L^2\)-Betti numbers in algebraic topology [M. F. Atiyah, Astérisque 32-33, 43-72 (1976; Zbl 0323.58015)]. The role of dimension in the definition of the Betti number is played by topological entropy. Results of Dodziuk, Cohen Cheeger and Gromov, Linnel, Lück have corresponding statements in this setup.
Reviewer: Manfred Denker (Göttingen)
Citations:
Zbl 0323.58015References:
| [1] | Atiyah, M. F., Elliptic operators, discrete groups and von Neumann algebras, Asterisque, 32-33, 43-72 (1976) · Zbl 0323.58015 |
| [2] | Cheeger, J.; Gromov, M., L^2-cohomology and group cohomology, Topology, 25, 189-215 (1986) · Zbl 0597.57020 · doi:10.1016/0040-9383(86)90039-X |
| [3] | Cohen, J. M., Von Neumann dimension and the homology of covering spaces, The Quarterly Journal of Mathematics. Oxford, 30, 133-142 (1979) · Zbl 0421.46056 · doi:10.1093/qmath/30.2.133 |
| [4] | Dodziuk, J., De Rham-Hodge theory for L^2-cohomology of infinite coverings, Topology, 16, 157-165 (1977) · Zbl 0348.58001 · doi:10.1016/0040-9383(77)90013-1 |
| [5] | Dodziuk, J.; Mathai, V., Approximating L^2-invariants of amenable covering spaces: A combinatorial approach, Journal of Functional Analysis, 154, 359-378 (1998) · Zbl 0936.57018 · doi:10.1006/jfan.1997.3205 |
| [6] | Elek, G., The Euler characteristic of discrete groups and Yuzvinskii’s entropy addition formula, The Bulletin of the London Mathematical Society, 31, 661-664 (1999) · Zbl 1017.37007 · doi:10.1112/S0024609399006104 |
| [7] | Farber, M., Geometry of growth: approximation theorems for L^2-invariants, Mathematische Annalen, 311, 335-375 (1998) · Zbl 0911.53026 · doi:10.1007/s002080050190 |
| [8] | Hewitt, E.; Ross, K. A., Abstract Harmonic Analysis II (1970), Berlin-Heidelberg-New York: Springer-Verlag, Berlin-Heidelberg-New York · Zbl 0213.40103 |
| [9] | Kitchens, B.; Schmidt, K., Automorphisms of compact groups, Ergodic Theory and Dynamical Systems, 9, 691-735 (1989) · Zbl 0709.54023 |
| [10] | Lind, D.; Schmidt, K.; Ward, T., Mahler measure and entropy for commuting automorphisms of compact groups, Inventiones Mathematicae, 101, 593-629 (1990) · Zbl 0774.22002 · doi:10.1007/BF01231517 |
| [11] | Linnell, P. A., Division rings and group von Neumann algebras, Forum Mathematicum, 5, 561-576 (1993) · Zbl 0794.22008 · doi:10.1515/form.1993.5.561 |
| [12] | Lück, W., Approximating L^2-invariants by their finite-dimensional analogues, Geometric and Functional Analysis, 4, 455-481 (1994) · Zbl 0853.57021 · doi:10.1007/BF01896404 |
| [13] | Lück, W., Dimension theory of arbitrary modules over finite von Neumann algebras and L^2-Betti numbers. II: Applications to Grothendieck groups, L^2-Euler characteristics and Burnside groups, Journal für die reine und angewandte Mathematik, 496, 213-236 (1998) · Zbl 1001.55019 · doi:10.1515/crll.1998.031 |
| [14] | Ollagnier, J. Moulin, Ergodic theory and statistical mechanics (1985), Berlin: Springer-Verlag, Berlin · Zbl 0558.28010 |
| [15] | Ornstein, D. S.; Weiss, B., Entropy and isomorphism theorems for actions of amenable groups, Journal d’Analyse Mathématique, 48, 1-141 (1987) · Zbl 0637.28015 · doi:10.1007/BF02790325 |
| [16] | Pansu, P., Introduction to L^2-Betti numbers, Fields Institute Monographs, 4, 53-86 (1996) · Zbl 0848.53025 |
| [17] | Passman, D. S., The Algebraic Structure of Group Rings (1977), New York: Wiley, New York · Zbl 0368.16003 |
| [18] | Rosenberg, J., Algebraic K-theory and its Applications (1994), Berlin: Springer-Verlag, Berlin · Zbl 0801.19001 |
| [19] | Ruelle, D., Thermodynamic formalism, Encyclopedia of Mathematics and Its Applications (1978), Reading, MA: Addison-Wesley, Reading, MA · Zbl 0401.28016 |
| [20] | Schmidt, K., Dynamical systems of algebraic origin (1995), Basel: Birkhäuser Verlag, Basel · Zbl 0833.28001 |
| [21] | Ward, T.; Zhang, Q., The Abramov-Rohlin entropy addition formula for amenable group actions, Monatshefte für Mathematik, 114, 317-329 (1992) · Zbl 0764.28014 · doi:10.1007/BF01299386 |
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