Entropy , Determinants , and L 2 - Torsion

We show that for any amenable group Γ and any ZΓ-module M of type FL with vanishing Euler characteristic, the entropy of the natural Γ-action on the Pontryagin dual of M is equal to the L-torsion of M. As a particular case, the entropy of the principal algebraic action associated with the module ZΓ/ZΓf is equal to the logarithm of the Fuglede-Kadison determinant of f whenever f is a non-zero-divisor in ZΓ. 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-torsion and entropy, we show that the L-torsion of a non-trivial amenable group with finite classifying space vanishes. This was conjectured by Lück. Finally, we establish a Milnor-Turaev formula for the L-torsion of a finite ∆-acyclic chain complex.