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Entropy, determinants, and \(L^{2}\)-torsion. (English) Zbl 1283.37031

Let \(\Gamma\) be a countable discrete amenable group. The entropy considered in this paper is the topological entropy of the action of \(\Gamma\) on a compact metrizable group by continuous automorphisms.
Denote by \(\mathbb Z\Gamma\) the integral group ring of \(\Gamma\), consider a \(\mathbb Z\Gamma\)-module \(M\) and the action of \(\Gamma\) on the Pontryagin dual \(\widehat M\) of \(M\). If \(M\) is of type FL and the Euler characteristic of \(M\) is zero, then the \(L^2\)-torsion of \(M\) is defined (in terms of the Fuglede-Kadison determinant, defined on the group von Neumann algebra). The main theorem of the paper states that, under these assumptions, the entropy of the action of \(\Gamma\) on \(\widehat M\) coincides with the \(L^2\)-torsion of \(M\). Several consequences of this equality between two numerical invariants of different nature are given for \(L^2\)-torsion and entropy.
An important consequence of the main theorem is the proof of a conjecture by Deninger. Indeed, the authors apply their main result when \(M\) is the quotient \(\mathbb Z\Gamma/\mathbb Z\Gamma f\), where \(f\) is a non-zero-divisor in \(\mathbb Z\Gamma\) and \(\mathbb Z\Gamma f\) is the principal ideal generated by \(f\) in \(\mathbb Z\Gamma\). In this case, the entropy of the natural action of \(\Gamma\) on \(\widehat M\) coincides with the logarithm of the Fuglede-Kadison determinant of \(f\). This extends classical results by Yuzvinski (for \(\Gamma=\mathbb Z\)) and by Lind, Schmidt and Ward (for \(\Gamma=\mathbb Z^d\)).
Another application of the main theorem confirms a conjecture by Lück. In fact, it is proved that the \(L^2\)-torsion of a non-trivial amenable group \(\Gamma\) is zero if the trivial \(\mathbb Z\Gamma\)-module \(\mathbb Z\) is of type FL.
Moreover, the authors generalize a result by Szegö to an approximation theorem for the Fuglede-Kadison determinant on the group von Neumann algebra of an amenable group, which is applied in the proof of the main theorem.
Finally, using the main theorem, the notion of torsion is introduced for a countable \(\mathbb Z\Gamma\)-module as the entropy of its Pontryagin dual, and it is given a Milnor-Turaev formula for the \(L^2\)-torsion of a finite \(\Delta\)-acyclic chain complex.

MSC:

37B40 Topological entropy
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
58J52 Determinants and determinant bundles, analytic torsion
43A07 Means on groups, semigroups, etc.; amenable groups
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[1] Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, 2nd ed., Graduate Texts in Mathematics, vol. 13, Springer-Verlag, New York, 1992. · Zbl 0765.16001
[2] Mladen Bestvina and Noel Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), no. 3, 445 – 470. · Zbl 0888.20021 · doi:10.1007/s002220050168
[3] Patrick Billingsley, Convergence of probability measures, 2nd ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. A Wiley-Interscience Publication. · Zbl 0944.60003
[4] Florin-Petre Boca, Rotation \?*-algebras and almost Mathieu operators, Theta Series in Advanced Mathematics, vol. 1, The Theta Foundation, Bucharest, 2001. · Zbl 1191.47001
[5] Lewis Bowen, Measure conjugacy invariants for actions of countable sofic groups, J. Amer. Math. Soc. 23 (2010), no. 1, 217 – 245. · Zbl 1201.37005
[6] Lewis Bowen, Entropy for expansive algebraic actions of residually finite groups, Ergodic Theory Dynam. Systems 31 (2011), no. 3, 703 – 718. · Zbl 1234.37010 · doi:10.1017/S0143385710000179
[7] Lewis Bowen and Hanfeng Li, Harmonic models and spanning forests of residually finite groups, J. Funct. Anal. 263 (2012), no. 7, 1769 – 1808. · Zbl 1271.37023 · doi:10.1016/j.jfa.2012.06.015
[8] Maxim Braverman, Alan Carey, Michael Farber, and Varghese Mathai, \?² torsion without the determinant class condition and extended \?² cohomology, Commun. Contemp. Math. 7 (2005), no. 4, 421 – 462. · Zbl 1079.55015 · doi:10.1142/S0219199705001866
[9] Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1994. Corrected reprint of the 1982 original.
[10] A. Carey, M. Farber, and V. Mathai, Determinant lines, von Neumann algebras and \?² torsion, J. Reine Angew. Math. 484 (1997), 153 – 181. · Zbl 0872.46031
[11] Varghese Mathai and Alan L. Carey, \?²-acyclicity and \?²-torsion invariants, Geometric and topological invariants of elliptic operators (Brunswick, ME, 1988) Contemp. Math., vol. 105, Amer. Math. Soc., Providence, RI, 1990, pp. 91 – 118. · doi:10.1090/conm/105/1047278
[12] Alan L. Carey and Varghese Mathai, \?²-torsion invariants, J. Funct. Anal. 110 (1992), no. 2, 377 – 409. · Zbl 0771.57009 · doi:10.1016/0022-1236(92)90036-I
[13] Tullio Ceccherini-Silberstein and Michel Coornaert, Cellular automata and groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. · Zbl 1218.37004
[14] N.-P. Chung and H. Li. Homoclinic groups, IE groups, and expansive algebraic actions. Preprint, 2011. · Zbl 1320.37009
[15] Alexandre I. Danilenko, Entropy theory from the orbital point of view, Monatsh. Math. 134 (2001), no. 2, 121 – 141. · Zbl 0996.37007 · doi:10.1007/s006050170003
[16] Marshall M. Cohen, A course in simple-homotopy theory, Springer-Verlag, New York-Berlin, 1973. Graduate Texts in Mathematics, Vol. 10. · Zbl 0261.57009
[17] Christopher Deninger, Deligne periods of mixed motives, \?-theory and the entropy of certain \?\(^{n}\)-actions, J. Amer. Math. Soc. 10 (1997), no. 2, 259 – 281. · Zbl 0913.11027
[18] Christopher Deninger, Fuglede-Kadison determinants and entropy for actions of discrete amenable groups, J. Amer. Math. Soc. 19 (2006), no. 3, 737 – 758. · Zbl 1104.22010
[19] Christopher Deninger, Mahler measures and Fuglede-Kadison determinants, Münster J. Math. 2 (2009), 45 – 63. · Zbl 1245.11107
[20] Christopher Deninger, Regulators, entropy and infinite determinants, Regulators, Contemp. Math., vol. 571, Amer. Math. Soc., Providence, RI, 2012, pp. 117 – 134. · Zbl 1318.19005 · doi:10.1090/conm/571/11324
[21] 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. · Zbl 1128.22003 · doi:10.1017/S0143385706000939
[22] Józef Dodziuk, Peter Linnell, Varghese Mathai, Thomas Schick, and Stuart Yates, Approximating \?²-invariants and the Atiyah conjecture, Comm. Pure Appl. Math. 56 (2003), no. 7, 839 – 873. Dedicated to the memory of Jürgen K. Moser. · Zbl 1036.58017 · doi:10.1002/cpa.10076
[23] Jozef Dodziuk and Varghese Mathai, Approximating \?² invariants of amenable covering spaces: a combinatorial approach, J. Funct. Anal. 154 (1998), no. 2, 359 – 378. · Zbl 0936.57018 · doi:10.1006/jfan.1997.3205
[24] Samuel Eilenberg and Tudor Ganea, On the Lusternik-Schnirelmann category of abstract groups, Ann. of Math. (2) 65 (1957), 517 – 518. · Zbl 0079.25401 · doi:10.2307/1970062
[25] Gábor Elek, On the analytic zero divisor conjecture of Linnell, Bull. London Math. Soc. 35 (2003), no. 2, 236 – 238. · Zbl 1027.20002 · doi:10.1112/S002460930200173X
[26] Gábor Elek and Endre Szabó, Hyperlinearity, essentially free actions and \?²-invariants. The sofic property, Math. Ann. 332 (2005), no. 2, 421 – 441. · Zbl 1070.43002 · doi:10.1007/s00208-005-0640-8
[27] Bent Fuglede and Richard V. Kadison, Determinant theory in finite factors, Ann. of Math. (2) 55 (1952), 520 – 530. · Zbl 0046.33604 · doi:10.2307/1969645
[28] F. P. Gantmacher and M. G. Krein, Oscillation matrices and kernels and small vibrations of mechanical systems, Revised edition, AMS Chelsea Publishing, Providence, RI, 2002. Translation based on the 1941 Russian original; Edited and with a preface by Alex Eremenko. · Zbl 1002.74002
[29] Ross Geoghegan, Topological methods in group theory, Graduate Texts in Mathematics, vol. 243, Springer, New York, 2008. · Zbl 1141.57001
[30] Uffe Haagerup and Hanne Schultz, Brown measures of unbounded operators affiliated with a finite von Neumann algebra, Math. Scand. 100 (2007), no. 2, 209 – 263. · Zbl 1168.46039 · doi:10.7146/math.scand.a-15023
[31] Henry Helson and David Lowdenslager, Prediction theory and Fourier series in several variables, Acta Math. 99 (1958), 165 – 202. · Zbl 0082.28201 · doi:10.1007/BF02392425
[32] Charles R. Johnson and Wayne W. Barrett, Spanning-tree extensions of the Hadamard-Fischer inequalities, Linear Algebra Appl. 66 (1985), 177 – 193. · Zbl 0619.05021 · doi:10.1016/0024-3795(85)90131-4
[33] Richard V. Kadison, Diagonalizing matrices, Amer. J. Math. 106 (1984), no. 6, 1451 – 1468. · Zbl 0585.46048 · doi:10.2307/2374400
[34] Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Graduate Studies in Mathematics, vol. 15, American Mathematical Society, Providence, RI, 1997. Elementary theory; Reprint of the 1983 original. Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II, Graduate Studies in Mathematics, vol. 16, American Mathematical Society, Providence, RI, 1997. Advanced theory; Corrected reprint of the 1986 original. · Zbl 0888.46039
[35] Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Graduate Studies in Mathematics, vol. 15, American Mathematical Society, Providence, RI, 1997. Elementary theory; Reprint of the 1983 original. Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II, Graduate Studies in Mathematics, vol. 16, American Mathematical Society, Providence, RI, 1997. Advanced theory; Corrected reprint of the 1986 original. · Zbl 0888.46039
[36] David Kerr and Hanfeng Li, Entropy and the variational principle for actions of sofic groups, Invent. Math. 186 (2011), no. 3, 501 – 558. · Zbl 1417.37041 · doi:10.1007/s00222-011-0324-9
[37] A. K. Kelmans and B. N. Kimel\(^{\prime}\)fel\(^{\prime}\)d, Multiplicative submodularity of a matrix’s principal minor as a function of the set of its rows, and some combinatorial applications, Discrete Math. 44 (1983), no. 1, 113 – 116. · Zbl 0524.15011 · doi:10.1016/0012-365X(83)90011-0
[38] P. H. Kropholler, C. Martinez-Pérez, and B. E. A. Nucinkis, Cohomological finiteness conditions for elementary amenable groups, J. Reine Angew. Math. 637 (2009), 49 – 62. · Zbl 1202.20055 · doi:10.1515/CRELLE.2009.090
[39] T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999. · Zbl 0911.16001
[40] Serge Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. · Zbl 0984.00001
[41] Hanfeng Li, Compact group automorphisms, addition formulas and Fuglede-Kadison determinants, Ann. of Math. (2) 176 (2012), no. 1, 303 – 347. · Zbl 1250.22006 · doi:10.4007/annals.2012.176.1.5
[42] Douglas Lind, Klaus Schmidt, and Evgeny Verbitskiy, Entropy and growth rate of periodic points of algebraic \Bbb Z^{\?}-actions, Dynamical numbers — interplay between dynamical systems and number theory, Contemp. Math., vol. 532, Amer. Math. Soc., Providence, RI, 2010, pp. 195 – 211. · Zbl 1217.54034 · doi:10.1090/conm/532/10491
[43] D. Lind, K. Schmidt, and E. Verbitskiy. Homoclinic points, atoral polynomials, and periodic points of algebraic \( \mathbb{Z}^d\)-actions. Ergod. Th. Dynam. Sys. 33 (2013), no. 4, 1060-1081. · Zbl 1294.37009
[44] 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. · Zbl 0774.22002 · doi:10.1007/BF01231517
[45] Elon Lindenstrauss and Benjamin Weiss, Mean topological dimension, Israel J. Math. 115 (2000), 1 – 24. · Zbl 0978.54026 · doi:10.1007/BF02810577
[46] I. Ju. Linnik, A multidimensional analogue of G. Szegő’s limit theorem, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 6, 1393 – 1403, 1439 (Russian).
[47] W. Lück, Approximating \?²-invariants by their finite-dimensional analogues, Geom. Funct. Anal. 4 (1994), no. 4, 455 – 481. · Zbl 0853.57021 · doi:10.1007/BF01896404
[48] Wolfgang Lück, \?²-invariants of regular coverings of compact manifolds and CW-complexes, Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 735 – 817. · Zbl 1069.57017
[49] Wolfgang Lück, \?²-invariants: theory and applications to geometry and \?-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. · Zbl 1009.55001
[50] Wolfgang Lück and Mikael Rørdam, Algebraic \?-theory of von Neumann algebras, \?-Theory 7 (1993), no. 6, 517 – 536. · Zbl 0802.19001 · doi:10.1007/BF00961216
[51] Wolfgang Lück and Mel Rothenberg, Reidemeister torsion and the \?-theory of von Neumann algebras, \?-Theory 5 (1991), no. 3, 213 – 264. · Zbl 0748.57007 · doi:10.1007/BF00533588
[52] Wolfgang Lück, Roman Sauer, and Christian Wegner, \?²-torsion, the measure-theoretic determinant conjecture, and uniform measure equivalence, J. Topol. Anal. 2 (2010), no. 2, 145 – 171. · Zbl 1195.57053 · doi:10.1142/S179352531000032X
[53] Russell Lyons, Asymptotic enumeration of spanning trees, Combin. Probab. Comput. 14 (2005), no. 4, 491 – 522. · Zbl 1076.05007 · doi:10.1017/S096354830500684X
[54] Russell Lyons, Identities and inequalities for tree entropy, Combin. Probab. Comput. 19 (2010), no. 2, 303 – 313. · Zbl 1215.05031 · doi:10.1017/S0963548309990605
[55] J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358 – 426. · Zbl 0147.23104
[56] John W. Milnor, Infinite cyclic coverings, Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967) Prindle, Weber & Schmidt, Boston, Mass., 1968, pp. 115 – 133.
[57] Jean Moulin Ollagnier, Ergodic theory and statistical mechanics, Lecture Notes in Mathematics, vol. 1115, Springer-Verlag, Berlin, 1985. · Zbl 0558.28010
[58] Donald S. Ornstein and Benjamin Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math. 48 (1987), 1 – 141. · Zbl 0637.28015 · doi:10.1007/BF02790325
[59] M. Scott Osborne, Basic homological algebra, Graduate Texts in Mathematics, vol. 196, Springer-Verlag, New York, 2000. · Zbl 0948.18001
[60] Donald S. Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. · Zbl 0368.16003
[61] Justin Peters, Entropy on discrete abelian groups, Adv. in Math. 33 (1979), no. 1, 1 – 13. · Zbl 0421.28019 · doi:10.1016/S0001-8708(79)80007-9
[62] Jesse Peterson and Andreas Thom, Group cocycles and the ring of affiliated operators, Invent. Math. 185 (2011), no. 3, 561 – 592. · Zbl 1227.22003 · doi:10.1007/s00222-011-0310-2
[63] Jonathan Rosenberg, Algebraic \?-theory and its applications, Graduate Texts in Mathematics, vol. 147, Springer-Verlag, New York, 1994. · Zbl 0801.19001
[64] Thomas Schick, \?²-determinant class and approximation of \?²-Betti numbers, Trans. Amer. Math. Soc. 353 (2001), no. 8, 3247 – 3265. · Zbl 0979.55004
[65] Klaus Schmidt, Dynamical systems of algebraic origin, Progress in Mathematics, vol. 128, Birkhäuser Verlag, Basel, 1995. · Zbl 0833.28001
[66] Barry Simon, Orthogonal polynomials on the unit circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Classical theory. Barry Simon, Orthogonal polynomials on the unit circle. Part 2, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Spectral theory. · Zbl 1082.42020
[67] Rita Solomyak, On coincidence of entropies for two classes of dynamical systems, Ergodic Theory Dynam. Systems 18 (1998), no. 3, 731 – 738. · Zbl 0924.58047 · doi:10.1017/S0143385798108313
[68] G. Szegö, Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion, Math. Ann. 76 (1915), no. 4, 490 – 503 (German). · JFM 45.0518.02 · doi:10.1007/BF01458220
[69] M. Takesaki, Theory of operator algebras. I, Encyclopaedia of Mathematical Sciences, vol. 124, Springer-Verlag, Berlin, 2002. Reprint of the first (1979) edition; Operator Algebras and Non-commutative Geometry, 5. · Zbl 0990.46034
[70] V. G. Turaev, Reidemeister torsion in knot theory, Uspekhi Mat. Nauk 41 (1986), no. 1(247), 97 – 147, 240 (Russian). · Zbl 0602.57005
[71] Vladimir Turaev, Torsions of 3-dimensional manifolds, Progress in Mathematics, vol. 208, Birkhäuser Verlag, Basel, 2002. · Zbl 1012.57034
[72] C. T. C. Wall, Finiteness conditions for \?\?-complexes, Ann. of Math. (2) 81 (1965), 56 – 69. · Zbl 0152.21902 · doi:10.2307/1970382
[73] C. T. C. Wall, Finiteness conditions for \?\? complexes. II, Proc. Roy. Soc. Ser. A 295 (1966), 129 – 139.
[74] Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. · Zbl 0475.28009
[75] Christian Wegner, \?²-invariants of finite aspherical CW-complexes, Manuscripta Math. 128 (2009), no. 4, 469 – 481. · Zbl 1163.57016 · doi:10.1007/s00229-008-0246-z
[76] S. A. Juzvinskiĭ, Calculation of the entropy of a group-endomorphism, Sibirsk. Mat. Ž. 8 (1967), 230 – 239 (Russian).
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