Entropy, determinants, and L^2-torsion

Hanfeng Li Chongqing University, SUNY at Buffalo Andreas Thom TU Dresden

Dynamical Systems Spectral Theory and Operator Algebra mathscidoc:1704.02001

Distinguished Paper Award in 2018

J. Amer. Math. Soc., 27, (1), 239--292, 2014
We show that for any amenable group \Gamma and any Z\Gamma-module M of type FL with vanishing Euler characteristic, the entropy of the natural \Gamma-action on the Pontryagin dual of M is equal to the L^2-torsion of M. As a particular case, the entropy of the principal algebraic action associated with the module Z\Gamma / Z\Gamma f is equal to the logarithm of the Fuglede-Kadison determinant of f whenever f is a non-zero-divisor in Z\Gamma. This confirms a conjecture of Deninger. As a key step in the proof we provide a general Szeg\H{o}-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 non-trivial amenable group with finite classifying space vanishes. This was conjectured by Lueck. Finally, we establish a Milnor-Turaev formula for the L^2-torsion of a finite \Delta-acyclic chain complex.
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  title={Entropy, determinants, and L^2-torsion},
  author={Hanfeng Li, and Andreas Thom},
  booktitle={J. Amer. Math. Soc.},
Hanfeng Li, and Andreas Thom. Entropy, determinants, and L^2-torsion. 2014. Vol. 27. In J. Amer. Math. Soc.. pp.239--292. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170430083031780722749.
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