Entropy, Determinants, and L2-Torsion

TL;DR

This paper establishes a deep connection between entropy and L2-torsion for amenable groups, confirming conjectures and providing new approximation theorems with implications for group theory and algebraic dynamics.

Contribution

It proves that entropy equals L2-torsion for certain modules over amenable groups, confirming Deninger's conjecture and introducing a Szegő-type approximation theorem.

Findings

Entropy equals L2-torsion for modules of type FL with vanishing Euler characteristic.

The entropy of algebraic actions equals the logarithm of the Fuglede-Kadison determinant for non-zero-divisors.

L2-torsion vanishes for non-trivial amenable groups with finite classifying space.

Abstract

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 L2-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 L2-torsion and entropy, we show that the L2-torsion of a non-trivial amenable group with finite classifying space vanishes. This was conjectured by L\"uck. Finally, we establish a Milnor-Turaev formula for the L2-torsion of a finite \Delta-acyclic chain complex.