Metric mean dimension for algebraic actions of Sofic groups

Abstract

Abstract. We prove that if Γ is a sofic group, and A is a finitely generated Z(Γ)-module, then the metric mean dimension of Γ A, in the sense of Hanfeng Li is equal to the von Neumann-Lück rank of A. This partially extends the results of Hanfeng Li and Bingbing Liang in [22] from the case of amenable groups to the case of sofic groups. Additionally we show that the mean dimension of Γ A is the von Neumann-Lück rank of A, if A is finitely presented and Γ is residually finite. It turns out that our approach naturally leads to a notion of p-metric mean dimension, which is in between mean dimension and the usual metric mean dimension. This can be seen as an obstruction to the equality of mean dimension and metric mean dimension. While we cannot decide if mean dimension is the same as metric mean dimension for algebraic actions, we show that in the metric case that for all p the p-metric mean dimension coincides with the von Neumann-Lück rank of the dual module.

Independence tuples and Deninger's problem

“…It is quite useful when one wishes to apply Hilbert space techniques as these are phrased better in terms of the ℓ 2 -product metric. This is precisely the purpose of their use in [16], [15], [17] and we believe this is crucial for those results, as well as the results in this paper. The second version of independence tuples is one in which we control the translates of an independence set J by the left regular representation (in a sense to be made more precise in Section 3), and moreover only consider partitions c : J → {1, .…”

“…are asymptotically the same notion. A crucial defect of the argument in [16] is that the proof of existence of φ is nonconstructive, using a compactness argument in an essential way. However, due to its nonconstructive nature it allows one to create more elements in Map(ρ, F, δ, σ i ) than one would initially believe exist.…”

“…We believe this is a very important technique, which often gives one added flexibility needed to prove results in entropy theory. We mention that we have already exploited this in [16], [15], [17]. It is quite useful when one wishes to apply Hilbert space techniques as these are phrased better in terms of the ℓ 2 -product metric.…”

Fuglede–Kadison Determinants and Sofic Entropy

“…the class of groups for which dynamical entropy is defined). Part (i) is a trivial consequence of the main results of our results in [19] (see Theorem 6.16 of this paper). Thus we focus on (ii),(iii) for most of the paper.…”

“…We shall present a Lemma from [19]. For terminology, if A ∈ M m,n (C), we use A 2 2 = tr n (A * A), we shall use A ∞ for the operator norm of A. Lemma 2.3 (Lemma 2.6 in [19]). Let Γ be a countable discrete sofic group wit sofic approximation Σ = (σ i : Γ → S di ).…”