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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
DOI: https://doi.org/10.1090/S0894-0347-2013-00778-X
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
Email: hfli@math.buffalo.edu

Andreas Thom
Affiliation: Mathematisches Institut, Universität Leipzig, PF 100920, 04009 Leipzig, Germany
MR Author ID: 780176
ORCID: 0000-0002-7245-2861
Email: thom@math.uni-leipzig.de

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.