Some remarks on the entropy for algebraic actions of amenable groups
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- by Nhan-Phu Chung and Andreas Thom PDF
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Abstract:
In this short note we study the entropy for algebraic actions of certain amenable groups. The possible values for this entropy are studied. Various fundamental results about certain classes of amenable groups are re-proved using elementary arguments and the entropy invariant. We provide a natural decomposition of the entropy into summands contributed by individual primes and a summand corresponding to $\infty$. These results extend previous work by Lind and Ward on $p$-adic entropy.References
- A. J. Berrick and J. A. Hillman, Perfect and acyclic subgroups of finitely presentable groups, J. London Math. Soc. (2) 68 (2003), no. 3, 683–698. MR 2009444, DOI 10.1112/S0024610703004587
- Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1994. Corrected reprint of the 1982 original. MR 1324339
- 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
- Ching Chou, Elementary amenable groups, Illinois J. Math. 24 (1980), no. 3, 396–407. MR 573475
- P. M. Cohn, Free ideal rings and localization in general rings, New Mathematical Monographs, vol. 3, Cambridge University Press, Cambridge, 2006. MR 2246388, DOI 10.1017/CBO9780511542794
- Christopher Deninger, Fuglede-Kadison determinants and entropy for actions of discrete amenable groups, J. Amer. Math. Soc. 19 (2006), no. 3, 737–758. MR 2220105, DOI 10.1090/S0894-0347-06-00519-4
- Christopher Deninger and Klaus Schmidt, Expansive algebraic actions of discrete residually finite amenable groups and their entropy, Ergodic Theory Dynam. Systems 27 (2007), no. 3, 769–786. MR 2322178, DOI 10.1017/S0143385706000939
- Manfred Einsiedler, A generalisation of Mahler measure and its application in algebraic dynamical systems, Acta Arith. 88 (1999), no. 1, 15–29. MR 1698350, DOI 10.4064/aa-88-1-15-29
- M. Einsiedler, G. Everest, and T. Ward, Entropy and the canonical height, J. Number Theory 91 (2001), no. 2, 256–273. MR 1876275, DOI 10.1006/jnth.2001.2682
- Gábor Elek, Amenable groups, topological entropy and Betti numbers, Israel J. Math. 132 (2002), 315–335. MR 1952628, DOI 10.1007/BF02784519
- Gábor Elek, The rank of finitely generated modules over group algebras, Proc. Amer. Math. Soc. 131 (2003), no. 11, 3477–3485. MR 1991759, DOI 10.1090/S0002-9939-03-06908-9
- G. R. Everest, On the elliptic analogue of Jensen’s formula, J. London Math. Soc. (2) 59 (1999), no. 1, 21–36. MR 1688486, DOI 10.1112/S0024610799007097
- G. R. Everest and Bríd Ní Fhlathúin, The elliptic Mahler measure, Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 1, 13–25. MR 1373343, DOI 10.1017/S0305004100074624
- Graham Everest and Thomas Ward, Heights of polynomials and entropy in algebraic dynamics, Universitext, Springer-Verlag London, Ltd., London, 1999. MR 1700272, DOI 10.1007/978-1-4471-3898-3
- Daniel R. Farkas and Robert L. Snider, $K_{0}$ and Noetherian group rings, J. Algebra 42 (1976), no. 1, 192–198. MR 422327, DOI 10.1016/0021-8693(76)90036-3
- Dion Gildenhuys, Classification of soluble groups of cohomological dimension two, Math. Z. 166 (1979), no. 1, 21–25. MR 526863, DOI 10.1007/BF01173844
- Misha Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps. I, Math. Phys. Anal. Geom. 2 (1999), no. 4, 323–415. MR 1742309, DOI 10.1023/A:1009841100168
- Jonathan A. Hillman, Elementary amenable groups and $4$-manifolds with Euler characteristic $0$, J. Austral. Math. Soc. Ser. A 50 (1991), no. 1, 160–170. MR 1094067, DOI 10.1017/S1446788700032638
- P. H. Kropholler, P. A. Linnell, and J. A. Moody, Applications of a new $K$-theoretic theorem to soluble group rings, Proc. Amer. Math. Soc. 104 (1988), no. 3, 675–684. MR 964842, DOI 10.1090/S0002-9939-1988-0964842-0
- T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999. MR 1653294, DOI 10.1007/978-1-4612-0525-8
- Wayne M. Lawton, A problem of Boyd concerning geometric means of polynomials, J. Number Theory 16 (1983), no. 3, 356–362. MR 707608, DOI 10.1016/0022-314X(83)90063-X
- Hanfeng Li, Compact group automorphisms, addition formulas and Fuglede-Kadison determinants, Ann. of Math. (2) 176 (2012), no. 1, 303–347. MR 2925385, DOI 10.4007/annals.2012.176.1.5
- Hanfeng Li and Bingbing Liang, Mean dimension, mean rank and von Neumann-Lück rank, arXiv preprint.
- Hanfeng Li and Andreas Thom, Entropy, determinants, and $L^2$-torsion, J. Amer. Math. Soc. 27 (2014), no. 1, 239–292. MR 3110799, DOI 10.1090/S0894-0347-2013-00778-X
- D. A. Lind and T. Ward, Automorphisms of solenoids and $p$-adic entropy, Ergodic Theory Dynam. Systems 8 (1988), no. 3, 411–419. MR 961739, DOI 10.1017/S0143385700004545
- Douglas Lind, Klaus Schmidt, and Tom Ward, Mahler measure and entropy for commuting automorphisms of compact groups, Invent. Math. 101 (1990), no. 3, 593–629. MR 1062797, DOI 10.1007/BF01231517
- Elon Lindenstrauss and Benjamin Weiss, Mean topological dimension, Israel J. Math. 115 (2000), 1–24. MR 1749670, DOI 10.1007/BF02810577
- 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
- Akio Noguchi, Zeros of the Alexander polynomial of knot, Osaka J. Math. 44 (2007), no. 3, 567–577. MR 2360941
- Donald S. Ornstein and Benjamin Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math. 48 (1987), 1–141. MR 910005, DOI 10.1007/BF02790325
- Donald S. Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR 0470211
- Justin Peters, Entropy on discrete abelian groups, Adv. in Math. 33 (1979), no. 1, 1–13. MR 540634, DOI 10.1016/S0001-8708(79)80007-9
- Thomas Schick, Integrality of $L^2$-Betti numbers, Math. Ann. 317 (2000), no. 4, 727–750. MR 1777117, DOI 10.1007/PL00004421
- Jean-Pierre Serre, Cohomologie des groupes discrets, Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970) Ann. of Math. Studies, No. 70, Princeton Univ. Press, Princeton, N.J., 1971, pp. 77–169 (French). MR 0385006
- Daniel S. Silver and Susan G. Williams, Torsion numbers of augmented groups with applications to knots and links, Enseign. Math. (2) 48 (2002), no. 3-4, 317–343. MR 1955605
- Dimitri Tamari, A refined classification of semi-groups leading to generalized polynomial rings with a generalized degree concept., Proceedings of the ICM, Amsterdam 3 (1954), 439–440.
- Sergej Yuzvinskiĭ, Calculation of the entropy of a group-endomorphism, Sibirsk. Mat. Z̆. 8 (1967), 230–239 (Russian).
Additional Information
- Nhan-Phu Chung
- Affiliation: Max Planck Institute for Math in the Sciences, Inselstraße 22, 04103 Leipzig, Germany
- Address at time of publication: Department of Mathematics, University of Sciences, Vietnam National University at Ho Chi Minh City, 227 Nguyen Van Cu, P4, Q5, TP.HCM, Vietnam
- MR Author ID: 962904
- Email: chung@mis.mpg.de, cnphu@hcmus.edu.vn
- Andreas Thom
- Affiliation: Mathematisches Institut, University of Leipzig, PF 100920, 04009 Leipzig, Germany
- Address at time of publication: Technische Universität Dresden, 01062 Dresden, Germany
- MR Author ID: 780176
- ORCID: 0000-0002-7245-2861
- Email: andreas.thom@math.uni-leipzig.de, andreas.thom@tu-dresden.de
- Received by editor(s): March 15, 2013
- Received by editor(s) in revised form: October 16, 2013
- Published electronically: November 12, 2014
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 8579-8595
- MSC (2010): Primary 37B40
- DOI: https://doi.org/10.1090/S0002-9947-2014-06348-4
- MathSciNet review: 3403066