ENTROPY, DETERMINANTS, AND L 2 -TORSION
By Hanfeng Li, Andreas Thom. 2012
59 citations recorded by the source corpus.
We show that for any amenable group and any Z-module M of type FL with vanishing Euler characteristic, the entropy of the natural -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 =Z f is equal to the logarithm of the Fuglede-Kadison determinant of f whenever f is a non-zero-divisor in Z. This conrms a conjecture of Deninger. As a key step in the proof we provide a general Szeg} 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 nite classifying space vanishes. This was conjectured by Luck. Finally, we establish a Milnor-Turaev formula for the L 2 -torsion of a nite -acyclic chain complex.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/401262372d1427519dfc92c46ff23cd17ba0cdd0
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Combinatorial Independence and Sofic Entropy (2012, unknown): “D S] 6 J an 2 01”
- Fuglede–Kadison Determinants and Sofic Entropy (2014, methodology): “e is a connection between entropy and Fuglede-Kadision determinants analogous to Theorem 1.1 for nonsofic groups, then the determinant conjecture should be true. As the proof of Theorem 1.1 as well as [34] rely on the fact that det+ L(Γ)(f) ≥ 1 for all f ∈ Mm,n(Z(Γ)), it is likely that any hypothetic”