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Entropy, determinants, and $ L^2$-torsion

Authors: Hanfeng Li and Andreas Thom
Journal: J. Amer. Math. Soc. 27 (2014), 239-292
MSC (2010): Primary 37B40, 37A35, 22D25, 58J52
Published electronically: July 23, 2013
MathSciNet review: 3110799
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Abstract: 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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Additional Information

Hanfeng Li
Affiliation: Department of Mathematics, Chongqing University, Chongqing 401331, China — and — Department of Mathematics, SUNY at Buffalo, Buffalo, New York 14260-2900

Andreas Thom
Affiliation: Mathematisches Institut, Universität Leipzig, PF 100920, 04009 Leipzig, Germany

Keywords: Entropy, amenable group, Fuglede-Kadison determinant, $L^2$-torsion
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.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.