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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Entropy, determinants, and $L^2$-torsion
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by Hanfeng Li and Andreas Thom
J. Amer. Math. Soc. 27 (2014), 239-292
Published electronically: July 23, 2013


We show that for any amenable group $\Gamma$ and any $\mathbb {Z} \Gamma$-module $\mathcal {M}$ of type FL with vanishing Euler characteristic, the entropy of the natural $\Gamma$-action on the Pontryagin dual of ${\mathcal {M}}$ is equal to the $L^2$-torsion of $\mathcal {M}$. As a particular case, the entropy of the principal algebraic action associated with the module $\mathbb {Z} \Gamma /\mathbb {Z} \Gamma f$ is equal to the logarithm of the Fuglede-Kadison determinant of $f$ whenever $f$ is a non-zero-divisor in $\mathbb {Z}\Gamma$. This confirms a conjecture of Deninger. As a key step in the proof we provide a general Szegő-type approximation theorem for the Fuglede-Kadison determinant on the group von Neumann algebra of an amenable group.

As a consequence of the equality between $L^2$-torsion and entropy, we show that the $L^2$-torsion of a nontrivial amenable group with finite classifying space vanishes. This was conjectured by Lück. Finally, we establish a Milnor-Turaev formula for the $L^2$-torsion of a finite $\Delta$-acyclic chain complex.

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Bibliographic Information
  • Hanfeng Li
  • Affiliation: Department of Mathematics, Chongqing University, Chongqing 401331, China — and — Department of Mathematics, SUNY at Buffalo, Buffalo, New York 14260-2900
  • Email:
  • Andreas Thom
  • Affiliation: Mathematisches Institut, Universität Leipzig, PF 100920, 04009 Leipzig, Germany
  • MR Author ID: 780176
  • ORCID: 0000-0002-7245-2861
  • Email:
  • Received by editor(s): June 1, 2012
  • Received by editor(s) in revised form: March 8, 2013
  • Published electronically: July 23, 2013
  • Additional Notes: The first author was partially supported by NSF Grants DMS-0701414 and DMS-1001625.
    The second author was supported by the ERC Starting Grant 277728.
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 27 (2014), 239-292
  • MSC (2010): Primary 37B40, 37A35, 22D25, 58J52
  • DOI:
  • MathSciNet review: 3110799