Conditional sofic mean dimension
Bingbing Liang
TL;DR
The paper develops a comprehensive framework for conditional mean dimension in the setting of sofic group actions, extending the amenable theory and connecting to Tsukamoto's relative mean dimension and Shi–Tsukamoto lower bounds. It introduces a reformulated width-dimension approach for conditional mean dimension, proves metric-independence under dynamical generation, and establishes two key reduction formulas that mirror the amenable case. The core results include a calculation framework for conditional sofic mean dimension on G-extensions, showing both lower and upper bounds that converge to equalities in natural regimes, and the metric version ${ m mdim_{ ext{Σ}}}$, including waist-type lower bounds extended to sofic actions. The work provides a robust set of tools to compare extension and fiberwise dimensions, relate conditional and relative dimensions, and raises an open question about the equivalence of two conditional notions in the sofic context, with potential implications for embedding problems and dimension theory in dynamical systems. Key constructs include: sofic approximations ${ m mdim_ ext{Σ}}$, Map$( ho_X,F, est, abla)$ spaces, G-extensions, and the relative notions ${ m mdim_ ext{Σ}}(X|Y)$ and ${ m mdim_ ext{Σ}}( ext{π})$.
Abstract
We undertake a study of the conditional mean dimensions for a factor map between continuous actions of a sofic group on two compact metrizable spaces. When the group is infinitely amenable, all these concepts recover as the conditional mean dimensions introduced in \cite{L22}. A range of results established for actions of amenable groups are extended to the sofic framework. Additionally, our exploration encompasses the study of the relative mean dimension introduced by Tsukamoto, shedding light on its inherent correlation with the conditional metric mean dimension within the sofic context. A lower bound on the conditional metric mean dimension, originally proposed by Shi-Tsukamoto, is extended to the sofic case.