Sofic Conditional Mean Dimension

• Sofic conditional mean dimension is a relative invariant that extends the amenable-group notion by quantifying asymptotic covering and embedding complexity in factor maps.
• It employs model space approximations, open-cover refinements, and metric variants to bridge classical mean dimension with sofic dynamics.
• The local tuple theory and zero-factor framework offer structural insights that aid in embedding, classification, and detecting positive versus zero extensions.

Sofic conditional mean dimension is a relative topological invariant for a factor map $\pi:(X,G)\to (Y,G)$ between continuous actions of a countable sofic group on compact metrizable spaces. It extends the amenable-group notion of conditional mean dimension to the sofic setting by measuring asymptotic covering or embedding complexity in model spaces, conditional on the factor. In the current literature, the subject includes the open-cover formulation of sofic conditional mean dimension, an embedding-dimension formulation, the companion notion of relative sofic mean dimension, metric variants, and a local theory based on tuples and sets that characterizes positivity and zero-factor structure (Liang, 2024, Li et al., 16 Aug 2025).

1. Amenable prototype and conceptual origin

The immediate antecedent is the amenable-group theory of conditional mean dimension for factor maps $\pi:X\to Y$. For a countable amenable group $\Gamma\curvearrowright X$, Bingbing Liang introduced the conditional order of a finite open cover $\mathcal U$ relative to $Y$ by

$$D(\mathcal{U}\mid Y) := \min \{ \operatorname{ord}(\mathcal{W}) \mid \mathcal{W} \text{ is a finite open cover of } X,~ \{\pi^{-1}(y)\}_{y \in Y} \vee \mathcal{W} \text{ refines } \mathcal{U} \},$$

and then defined

$$\operatorname{mdim}(X \mid Y) := \sup_{\mathcal{U}} \lim_{F} \frac{D(\mathcal{U}^F\mid Y)}{|F|}.$$

This reduces to the usual mean topological dimension when $Y$ is a singleton, and to topological dimension when $\Gamma$ is trivial (Liang, 2020).

The amenable theory established the dynamical analogue of Hurewicz-type dimension inequalities. In particular,

$$\operatorname{mdim}(X) \leq \operatorname{mdim}(Y) + \operatorname{mdim}(X \mid Y),$$

and it related the conditional invariant to fiberwise mean dimensions through

$\pi:X\to Y$0

It also computed the invariant for $\pi:X\to Y$1-extensions and connected it to embedding problems via Rokhlin dimension criteria (Liang, 2020).

This amenable framework supplied the formal template for the sofic theory. The 2020 paper explicitly noted that sofic analogues were not the main object there, but that the conditional notion could serve as a model for a factor-relative sofic mean dimension theory. The later sofic literature realizes precisely that program (Liang, 2020).

2. Definitions in the sofic setting

Sofic conditional mean dimension is built on Hanfeng Li’s sofic mean dimension formalism. A countable group $\pi:X\to Y$2 is sofic if it admits a sofic approximation sequence $\pi:X\to Y$3 satisfying asymptotic multiplicativity, asymptotic freeness, and $\pi:X\to Y$4. For a continuous action $\pi:X\to Y$5, Li defined model spaces $\pi:X\to Y$6 of approximately equivariant maps and normalized covering-dimension growth on these spaces, producing $\pi:X\to Y$7, independent of the choice of compatible metric (Li, 2011).

For a factor map $\pi:X\to Y$8, the embedding-dimension formulation of conditional sofic mean dimension begins with the model set

$\pi:X\to Y$9

where $\Gamma\curvearrowright X$0 is a compatible or dynamically generating pseudometric. If $\Gamma\curvearrowright X$1, an $\Gamma\curvearrowright X$2-embedding is a continuous map $\Gamma\curvearrowright X$3 such that equality of images together with $\Gamma\curvearrowright X$4-closeness of $\Gamma\curvearrowright X$5-images implies $\Gamma\curvearrowright X$6-closeness in $\Gamma\curvearrowright X$7. The minimal covering dimension of a target $\Gamma\curvearrowright X$8 admitting such an embedding is denoted $\Gamma\curvearrowright X$9, and the conditional invariant is defined by

$\mathcal U$0

For compatible metrics this is independent of the metric choices, and one writes $\mathcal U$1 (Liang, 2024).

A complementary open-cover formulation was later developed. For a finite open cover $\mathcal U$2 of $\mathcal U$3,

$\mathcal U$4

and

$\mathcal U$5

The 2025 work states that this equals the value given by Liang in terms of minimal dimension of compact metric spaces admitting relative embeddings, thereby unifying open-cover and embedding perspectives (Li et al., 16 Aug 2025).

3. Relative and metric variants

Alongside conditional sofic mean dimension, the literature studies relative sofic mean dimension, described in the 2024 paper as the sofic counterpart of the relative mean dimension introduced by Tsukamoto. In that formulation, one examines the mean dimension of fibers of $\mathcal U$6 inside model spaces, uniformly over $\mathcal U$7, obtaining an invariant denoted $\mathcal U$8 (Liang, 2024).

There is also a metric theory. For the conditional setting, one defines $\mathcal U$9 as the maximal cardinality of an $Y$0-separated subset of $Y$1, then sets

$Y$2

and

$Y$3

For compatible metrics this yields $Y$4 (Liang, 2024).

The 2024 paper emphasizes two points about these variants. First, when $Y$5 is amenable, the sofic definitions recover the classical conditional mean dimensions from the amenable theory. Second, the study includes the inherent correlation between relative mean dimension and conditional metric mean dimension in the sofic context, and extends to sofic groups a lower bound on conditional metric mean dimension originally proposed by Shi–Tsukamoto (Liang, 2024).

4. Structural properties and zero-factor theory

The sofic theory inherits several formal properties familiar from both the amenable case and the absolute sofic case. In the open-cover formulation, refinement is monotone: if $Y$6, then $Y$7. The invariant behaves appropriately under composition with factor maps and isomorphisms, and one has the product estimate

$Y$8

For each finite open cover $Y$9,

$$D(\mathcal{U}\mid Y) := \min \{ \operatorname{ord}(\mathcal{W}) \mid \mathcal{W} \text{ is a finite open cover of } X,~ \{\pi^{-1}(y)\}_{y \in Y} \vee \mathcal{W} \text{ refines } \mathcal{U} \},$$0

For infinite amenable groups, the product formula specializes to

$$D(\mathcal{U}\mid Y) := \min \{ \operatorname{ord}(\mathcal{W}) \mid \mathcal{W} \text{ is a finite open cover of } X,~ \{\pi^{-1}(y)\}_{y \in Y} \vee \mathcal{W} \text{ refines } \mathcal{U} \},$$1

described in the 2025 paper as generalizing the Jin–Qiao product formula to the conditional case (Li et al., 16 Aug 2025).

A major structural result concerns zero factors. A factor map $$D(\mathcal{U}\mid Y) := \min \{ \operatorname{ord}(\mathcal{W}) \mid \mathcal{W} \text{ is a finite open cover of } X,~ \{\pi^{-1}(y)\}_{y \in Y} \vee \mathcal{W} \text{ refines } \mathcal{U} \},$$2 is called a zero sofic conditional mean dimension extension when $$D(\mathcal{U}\mid Y) := \min \{ \operatorname{ord}(\mathcal{W}) \mid \mathcal{W} \text{ is a finite open cover of } X,~ \{\pi^{-1}(y)\}_{y \in Y} \vee \mathcal{W} \text{ refines } \mathcal{U} \},$$3. If $$D(\mathcal{U}\mid Y) := \min \{ \operatorname{ord}(\mathcal{W}) \mid \mathcal{W} \text{ is a finite open cover of } X,~ \{\pi^{-1}(y)\}_{y \in Y} \vee \mathcal{W} \text{ refines } \mathcal{U} \},$$4, then there exists a maximal zero sofic conditional mean dimension factor $$D(\mathcal{U}\mid Y) := \min \{ \operatorname{ord}(\mathcal{W}) \mid \mathcal{W} \text{ is a finite open cover of } X,~ \{\pi^{-1}(y)\}_{y \in Y} \vee \mathcal{W} \text{ refines } \mathcal{U} \},$$5 such that $$D(\mathcal{U}\mid Y) := \min \{ \operatorname{ord}(\mathcal{W}) \mid \mathcal{W} \text{ is a finite open cover of } X,~ \{\pi^{-1}(y)\}_{y \in Y} \vee \mathcal{W} \text{ refines } \mathcal{U} \},$$6 is a factor of $$D(\mathcal{U}\mid Y) := \min \{ \operatorname{ord}(\mathcal{W}) \mid \mathcal{W} \text{ is a finite open cover of } X,~ \{\pi^{-1}(y)\}_{y \in Y} \vee \mathcal{W} \text{ refines } \mathcal{U} \},$$7, $$D(\mathcal{U}\mid Y) := \min \{ \operatorname{ord}(\mathcal{W}) \mid \mathcal{W} \text{ is a finite open cover of } X,~ \{\pi^{-1}(y)\}_{y \in Y} \vee \mathcal{W} \text{ refines } \mathcal{U} \},$$8, and every other zero sofic conditional mean dimension factor is a factor of $$D(\mathcal{U}\mid Y) := \min \{ \operatorname{ord}(\mathcal{W}) \mid \mathcal{W} \text{ is a finite open cover of } X,~ \{\pi^{-1}(y)\}_{y \in Y} \vee \mathcal{W} \text{ refines } \mathcal{U} \},$$9. The same paper states the analogous result for relative sofic mean dimension. It presents this as a generalization of the Lindenstrauss–Weiss universal zero mean dimension factor to the conditional and relative sofic setting (Li et al., 16 Aug 2025).

This zero-factor theory situates sofic conditional mean dimension within a factor-sensitive hierarchy: non-negativity ensures existence of a universal zero object, while positivity excludes collapse to that maximal zero quotient. A plausible implication is that the conditional invariant functions as a structural separator among intermediate extensions, in the same way that relative entropy and zero-entropy factors organize factor structure in entropy theory.

5. Local theory: tuples, positivity, and complete positivity

A distinctive development in the sofic setting is the localization of conditional mean dimension. For $$\operatorname{mdim}(X \mid Y) := \sup_{\mathcal{U}} \lim_{F} \frac{D(\mathcal{U}^F\mid Y)}{|F|}.$$0, a tuple $$\operatorname{mdim}(X \mid Y) := \sup_{\mathcal{U}} \lim_{F} \frac{D(\mathcal{U}^F\mid Y)}{|F|}.$$1 is a sofic conditional mean dimension tuple relevant to $$\operatorname{mdim}(X \mid Y) := \sup_{\mathcal{U}} \lim_{F} \frac{D(\mathcal{U}^F\mid Y)}{|F|}.$$2 and $$\operatorname{mdim}(X \mid Y) := \sup_{\mathcal{U}} \lim_{F} \frac{D(\mathcal{U}^F\mid Y)}{|F|}.$$3 if for every open cover that does not simultaneously contain all $$\operatorname{mdim}(X \mid Y) := \sup_{\mathcal{U}} \lim_{F} \frac{D(\mathcal{U}^F\mid Y)}{|F|}.$$4 in any closed element, the associated sofic conditional mean dimension is positive. The set of such tuples is denoted

$$\operatorname{mdim}(X \mid Y) := \sup_{\mathcal{U}} \lim_{F} \frac{D(\mathcal{U}^F\mid Y)}{|F|}.$$5

The central characterization is

$$\operatorname{mdim}(X \mid Y) := \sup_{\mathcal{U}} \lim_{F} \frac{D(\mathcal{U}^F\mid Y)}{|F|}.$$6

Thus positive sofic conditional mean dimension is equivalent to the existence of a local witness tuple (Li et al., 16 Aug 2025).

The local sets satisfy several regularity properties. The set $$\operatorname{mdim}(X \mid Y) := \sup_{\mathcal{U}} \lim_{F} \frac{D(\mathcal{U}^F\mid Y)}{|F|}.$$7 is closed. Moreover,

$$\operatorname{mdim}(X \mid Y) := \sup_{\mathcal{U}} \lim_{F} \frac{D(\mathcal{U}^F\mid Y)}{|F|}.$$8

so tuples must lie in a common fiber and be non-diagonal. The same framework extends from tuples to subsets $$\operatorname{mdim}(X \mid Y) := \sup_{\mathcal{U}} \lim_{F} \frac{D(\mathcal{U}^F\mid Y)}{|F|}.$$9: such a set is a sofic conditional mean dimension set precisely when all tuples of distinct points in $Y$0 are conditional mean dimension tuples (Li et al., 16 Aug 2025).

The 2025 paper also defines sofic $Y$1-CPCMD and sofic $Y$2-UPCMD. Here CPCMD requires that every nontrivial intermediate factor $Y$3 have positive conditional mean dimension, while UPCMD requires positivity for every non-dense-on-fiber cover with $Y$4 sets. The characterization states that $Y$5 has sofic $Y$6-CPCMD if and only if the smallest closed $Y$7-invariant relation generated by $Y$8 is $Y$9, and uniform positivity implies complete positivity (Li et al., 16 Aug 2025).

This local theory has a clear antecedent in the absolute sofic theory of mean dimension pairs. There, a pair $\Gamma$0 is a sofic mean dimension pair if every standard two-set open cover distinguishing the pair has positive sofic mean dimension, UPMD holds exactly when every off-diagonal pair is such a pair, and a closure criterion for the relation generated by these pairs yields complete positivity. That earlier work explicitly described its local theory as a step toward defining a conditional mean dimension for sofic group actions (García-Ramos et al., 2024).

6. Relation to neighboring invariants and broader significance

Sofic conditional mean dimension belongs to a broader landscape of sofic mean-dimension-type invariants. At the absolute level, sofic mean dimension was introduced to generalize Gromov–Lindenstrauss–Weiss mean dimension from amenable to countable sofic groups and to distinguish actions with infinite entropy (Li, 2011). The conditional theory inherits that purpose in factor-relative form: it measures complexity not of the whole action in isolation, but of the extension over a specified factor (Liang, 2024).

The neighboring invariant most explicitly compared with sofic mean dimension in recent work is naive mean dimension. For continuous actions of any countable group on compact metrizable spaces, naive mean dimension is defined from $\Gamma$1, is independent of the compatible metric, and coincides with classical mean dimension for amenable groups. For sofic groups, the 2024 paper shows that naive mean dimension serves as an upper bound of sofic mean dimension for actions of nonamenable groups, and for algebraic actions it yields the comparison

$\Gamma$2

This suggests a surrounding hierarchy in which conditional sofic mean dimension should be read as one member of a family of increasingly refined sofic invariants (Liang et al., 2024).

There is also an algebraic analogue of relative complexity in sofic mean length. For a sofic group $\Gamma$3, a length function $\Gamma$4, and locally $\Gamma$5-finite $\Gamma$6-modules, the relative mean length $\Gamma$7 satisfies an addition formula

$\Gamma$8

The summary of that work explicitly states that this relative structure mirrors the relative mean topological or metric dimension for dynamical systems with a factor map (Li et al., 2015).

Within dynamics proper, the amenable conditional theory already showed that conditional mean dimension supports embedding applications. If a factor $\Gamma$9 has finite Rokhlin dimension $$\operatorname{mdim}(X) \leq \operatorname{mdim}(Y) + \operatorname{mdim}(X \mid Y),$$0 and $$\operatorname{mdim}(X) \leq \operatorname{mdim}(Y) + \operatorname{mdim}(X \mid Y),$$1, then $$\operatorname{mdim}(X) \leq \operatorname{mdim}(Y) + \operatorname{mdim}(X \mid Y),$$2 embeds equivariantly into $$\operatorname{mdim}(X) \leq \operatorname{mdim}(Y) + \operatorname{mdim}(X \mid Y),$$3, and analogous criteria were obtained for $$\operatorname{mdim}(X) \leq \operatorname{mdim}(Y) + \operatorname{mdim}(X \mid Y),$$4-extensions with continuous section (Liang, 2020). The sofic papers do not present the same embedding theorem in the supplied summaries, but they do describe the local and zero-factor machinery as a framework for structure theory and for applications such as embedding problems, classification, and the detection of “irreducible” complexity in extensions (Li et al., 16 Aug 2025).

In this sense, sofic conditional mean dimension is both a direct extension of the amenable relative invariant and a node in a larger network of sofic topological and algebraic invariants. Its present form combines model-space asymptotics, factor-relative covering or embedding dimension, and local tuple theory into a unified framework for studying extensions of sofic dynamical systems.