[Paper Review] Metric Mean Dimension for Algebraic Actions of Sofic Groups

Ben Hayes|arXiv (Cornell University)|Oct 15, 2013

Advanced Topology and Set Theory20 references3 citations

TL;DR

This paper establishes that for sofic groups Γ and finitely generated ℤ[Γ]-modules A, the metric mean dimension of the algebraic action Γ↷Â equals the von Neumann–Lück rank of A. The proof leverages microstate techniques and introduces p-metric mean dimension, showing it coincides with the von Neumann–Lück rank for all p, offering a potential obstruction to mean dimension and metric mean dimension being equal in general.

ABSTRACT

Recently Bingbing Liang and Hanfeng Li computed the mean dimension and metric mean dimension for algebraic actions of amenable groups. We show how to extend their computation of metric mean dimension to the case of sofic groups, provided that the dual module is finitely generated. Additionally, we show that when the dual module is finitely presented that the mean dimension is the von Neumann rank. The proof also goes through introducing \ell^{p}-analogues of metric mean dimension, which may be seen as an obstruction to the equality of mean dimension and metric mean dimension.

Motivation & Objective

• To extend the equality of metric mean dimension and von Neumann–Lück rank from amenable to sofic groups.
• To investigate whether mean dimension and metric mean dimension coincide for algebraic actions of sofic groups.
• To introduce and analyze p-metric mean dimension as a refinement of metric mean dimension.
• To explore the implications of p-metric mean dimension for the additivity and invariance of mean dimension in algebraic dynamics.
• To address the open question of whether mean dimension equals metric mean dimension by analyzing p-metric variants.

Proposed method

• The proof uses microstate techniques to approximate the dual module  in the topology of pointwise convergence, relying on compactness arguments to construct ε-almost equivariant maps.
• A key step involves showing that for any continuous pseudometric ρ on (𝕋ⁿ)Γ, a certain neighborhood N of the kernel of the action is ε-close to  in ρ, using a compactness-based construction.
• The authors define p-metric mean dimension for p ∈ [1, ∞) as a generalization of metric mean dimension, using ℓp-norms to measure distances in microstate sets.
• The von Neumann–Lück rank is computed via L²-invariants and trace in the group von Neumann algebra, which allows the use of Hilbert space techniques.
• The proof relies on the relative mean dimension framework and the use of sofic approximations to define asymptotic invariants.
• The construction of p-metric mean dimension is shown to be bounded below by the usual mean dimension, forming a chain of inequalities.

Experimental results

Research questions

• RQ1 Does the equality of metric mean dimension and von Neumann–Lück rank extend from amenable to sofic groups for finitely generated ℤ[Γ]-modules?
• RQ2 Can the p-metric mean dimension of Γ↷Â be shown to equal the von Neumann–Lück rank of A for all p ∈ [1, ∞)?
• RQ3 Is there a difference between p-metric mean dimension and q-metric mean dimension for p ≠ q, and if so, does this imply that mean dimension and metric mean dimension are not equal?
• RQ4 Can the finite generation assumption in the main theorem be removed to cover all ℤ[Γ]-modules?
• RQ5 Is it possible to construct a continuous ε-close map from a microstate set to Â, thereby enabling a stronger proof of mean dimension equality?

Key findings

• For any sofic group Γ and finitely generated ℤ[Γ]-module A, the metric mean dimension of the algebraic action Γ↷Â is equal to the von Neumann–Lück rank of A.
• When A is finitely presented and Γ is residually finite with sofic approximation from finite quotients, the mean dimension of Γ↷Â equals the von Neumann–Lück rank of A.
• The p-metric mean dimension is well-defined for all p ∈ [1, ∞), and for algebraic actions, it coincides with the von Neumann–Lück rank of the dual module.
• The p-metric mean dimension forms a nested chain of inequalities, with mean dimension ≤ p-metric mean dimension ≤ metric mean dimension for all p.
• The construction of p-metric mean dimension provides a potential obstruction to the equality of mean dimension and metric mean dimension, should they differ for different p.
• The paper shows that the von Neumann–Lück rank is an invariant of the topological dynamical system Γ↷Â for sofic groups, extending previous results that held only for amenable groups.