Fuglede-Kadison Determinants and Sofic Entropy

Abstract

We relate Fuglede-Kadison determinants to entropy of algebraic actions of sofic groups in essentially complete generality. This generalizes recent results of Hanfeng Li and Andreas Thom from the amenable case to the sofic case, as well as results of David Kerr, Hanfeng Li, and Lewis Bowen in the residually finite case. Moreover, the proof given is the first calculation of entropy of algebraic actions to avoid a nontrivial determinant approximation. We apply our results to a problem of Christopher Deninger, generalizing results of Hanfeng Li and David Kerr. Finally, we show that in many cases, finiteness of topological entropy for algebraic actions of sofic groups is equivalent to having metric mean dimension zero.

56 pages. This is the final version

References in corpus (4)

• Polish Models and Sofic Entropy
• Group ring elements with large spectral density
• Mean Dimension, Mean Rank, and von Neumann-Lück Rank
• Metric Mean Dimension for Algebraic Actions of Sofic Groups

Cited by in corpus (5)

• Examples in the entropy theory of countable group actions
• Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group
• Sofic Entropy of Gaussian Actions
• New Examples of Bernoulli Algebraic Actions
• Algebraic dynamical systems from LDPC codes satisfy a strong negation of the weak Pinsker property