Entropy, determinants, and 𝐿²-torsion

Li, Hanfeng; Thom, Andreas

doi:10.1090/S0894-0347-2013-00778-X

Entropy, determinants, and $L^2$-torsion
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by Hanfeng Li and Andreas Thom

J. Amer. Math. Soc. 27 (2014), 239-292

DOI: https://doi.org/10.1090/S0894-0347-2013-00778-X

Published electronically: July 23, 2013

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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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