Fuglede–Kadison Determinants and Sofic Entropy

By Ben Hayes. 2014

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We relate Fuglede–Kadison determinants to entropy of finitely-presented algebraic actions in essentially complete generality. We show that if $${f\in M_{m,n}(\mathbb{Z}(\Gamma))}$$f∈Mm,n(Z(Γ)) is injective as a left multiplication operator on $${\ell^{2}(\Gamma)^{\oplus n},}$$ℓ2(Γ)⊕n, then the topological entropy of the action of $${\Gamma}$$Γ on the dual of $${\mathbb{Z}(\Gamma)^{\oplus n}/\mathbb{Z}(\Gamma)^{\oplus m}f}$$Z(Γ)⊕n/Z(Γ)⊕mf is at most the logarithm of the positive Fuglede–Kadison determinant of f, with equality if m = n. We also prove that when m = n the measure-theoretic entropy of the action of $${\Gamma}$$Γ on the dual of $${\mathbb{Z}(\Gamma)^{\oplus n}/\mathbb{Z}(\Gamma)^{\oplus n}f}$$Z(Γ)⊕n/Z(Γ)⊕nf is the logarithm of the Fuglede–Kadison determinant of f. This work completely settles the connection between entropy of principal algebraic actions and Fuglede–Kadison determinants in the generality in which dynamical entropy is defined. Our main Theorem partially generalizes results of Li-Thom from amenable groups to sofic groups. Moreover, we show that the obvious full generalization of the Li-Thom theorem for amenable groups is false for general sofic groups. Lastly, we undertake a study of when the Yuzvinskiǐ addition formula fails for a non-amenable sofic group $${\Gamma}$$Γ, showing it always fails if $${\Gamma}$$Γ contains a nonabelian free group, and relating it to the possible values of L2-torsion in general.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0167748d3a4b0de0c0cf45477aa0532934de33e2