Relative Sofic Mean Dimension

• Relative sofic mean dimension is a factor-sensitive invariant for continuous actions of countable sofic groups on compact metrizable spaces, capturing the geometric complexity of factor maps.
• It differentiates between fiberwise and image-model approaches by evaluating the normalized width dimension of approximately equivariant model spaces.
• The framework extends classical amenable theories and provides precise connections between dynamical, metric, and algebraic invariants in topological dynamics.

Relative sofic mean dimension is a family of factor-sensitive invariants for continuous actions of countable sofic groups on compact metrizable spaces. In the current literature, the term is not completely uniform: one line of work uses it for a fiberwise sofic analogue of Tsukamoto’s relative mean dimension attached to a factor map $\pi:X\to Y$, while another uses it for the dimension growth of downstairs model images $$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$ seen through an extension $X\to Y$. Both viewpoints are built from sofic model spaces and normalized width dimension, but they measure different geometric aspects of the same factor map (Liang, 2024, Li et al., 16 Aug 2025).

1. Absolute sofic mean dimension as the ambient framework

Relative theories inherit their basic architecture from Hanfeng Li’s definition of sofic mean dimension. One fixes a countable sofic group $G$, a sofic approximation sequence

$$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$

a compact metrizable $G$-space $X$, and a continuous pseudometric $\rho$ on $X$. For finite $F\subset G$, $$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$0, and $$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$1, the model space is

$$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$2

where $$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$3. Sofic mean dimension is then defined from normalized width dimension of these approximately equivariant model spaces, while sofic metric mean dimension is defined from normalized separated or spanning growth in the same spaces (Li, 2011, Liang, 2024).

For factor maps $$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$4, the central operation is to compare upstairs model spaces, downstairs model spaces, and fibers of the induced coordinate map

$$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$5

All relative and conditional notions in the later literature arise by replacing the absolute model-space complexity of $$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$6 with some factor-sensitive complexity of $$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$7 or of its fibers (Liang, 2024, Li et al., 16 Aug 2025).

2. Two principal meanings of “relative sofic mean dimension”

The terminology now splits into several related invariants.

Notion in the literature	Defining object	Role
$$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$8 (Liang, 2024)	Exact fibers inside upstairs model spaces	Sofic analogue of Tsukamoto’s relative mean dimension
$$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$9 (Liang, 2024)	Whole upstairs model space with fiber-compatible embeddings	Conditional sofic mean dimension
$X\to Y$0 (Li et al., 16 Aug 2025)	Downstairs image model spaces $X\to Y$1	Relative complexity of the factor through the extension

In Liang’s 2024 formulation, the fiberwise invariant attached to a factor map $X\to Y$2 is

$X\to Y$3

This is explicitly presented as the sofic analogue of Tsukamoto’s relative mean dimension and measures the asymptotic width dimension of the worst exact fiber in the model space (Liang, 2024).

The same paper also defines conditional sofic mean dimension. Here one does not isolate a single fiber. Instead, one studies embeddings of the full model space $X\to Y$4 that separate points only when they lie in the same or approximately the same factor fiber. For $X\to Y$5,

$X\to Y$6

and then one takes infima in $X\to Y$7 and a supremum in $X\to Y$8 to obtain $X\to Y$9 (Liang, 2024).

A different usage appears in the 2025 paper on “Sofic conditional mean dimension, relative sofic mean dimension and their localizations.” There the relative invariant of the factor $G$0 with respect to the extension $G$1 is built from the image model spaces

$G$2

For a finite open cover $G$3 of $G$4,

$G$5

$G$6

and

$G$7

The same paper recalls an equivalent width-dimension formulation based on

$G$8

so the invariant again measures normalized width dimension, but now downstairs rather than fiberwise upstairs (Li et al., 16 Aug 2025).

3. Comparison theorems and extension formulas

The first basic comparison is that Liang’s fiberwise relative invariant is bounded by the conditional one: $G$9 This is formalized in Proposition 4.12 of the 2024 paper and expresses that worst-fiber complexity is dominated by the complexity of simultaneously controlling all fibers (Liang, 2024).

For amenable groups, both conditional and fiberwise sofic notions recover their classical predecessors. If $$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$0 is an infinite amenable group, then

$$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$1

and

$$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$2

Thus the sofic constructions are genuine extensions of amenable conditional mean dimension and Tsukamoto’s relative mean dimension rather than unrelated analogues (Liang, 2024).

The same paper establishes a factor-sensitive extension inequality

$$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$3

Its form reflects the fact that sofic mean dimension does not admit the naive factor monotonicity pattern one expects in the amenable category; the correction term is itself relative (Liang, 2024).

Several model computations clarify the geometry. For a product system $$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$4 with the projection $$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$5,

$$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$6

and if $$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$7, then

$$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$8

For a $$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$9-extension $G$0, if $G$1, then

$G$2

An explicit example is the Ornstein–Weiss factor map for the free group, where the kernel is isomorphic to $G$3; the paper concludes that the resulting relative sofic mean dimension is $G$4 (Liang, 2024).

On the metric side, Liang proves that conditional sofic metric mean dimension can be defined either with approximate fibers or with exact fibers: $G$5 He then shows that the fiberwise relative invariant is bounded above by the conditional metric one: $G$6 In particular, if $G$7, then

$G$8

This is a direct relative mean-dimension analogue of the standard entropy-versus-metric-mean-dimension principle (Liang, 2024).

4. Localization, positivity, and maximal zero factors

The 2025 localization paper develops a full local theory for the image-model invariant $G$9. At the cover level it proves an upper bound by the cover dimension, monotonicity under refinement, subadditivity under joins, isomorphism invariance downstairs, monotonicity through intermediate factors, subsystem monotonicity, and product inequalities. A technical subtlety is that these invariants may equal $X$0: if

$X$1

then by convention

$X$2

Accordingly, the non-negativity assumption in the zero-factor theory is substantive rather than cosmetic (Li et al., 16 Aug 2025).

Under the hypothesis

$X$3

the paper proves the existence of a maximal relative zero sofic mean dimension factor

$X$4

This factor is universal among all intermediate factors $X$5 over $X$6 with zero relative sofic mean dimension over $X$7: every such $X$8 factors through $X$9 (Li et al., 16 Aug 2025).

The same work localizes positivity via tuples. A tuple $\rho$0 is called a relative sofic mean dimension tuple if every admissible open cover $\rho$1 with respect to $\rho$2 satisfies

$\rho$3

Writing

$\rho$4

for the set of such tuples, the main characterization is

$\rho$5

for some $\rho$6. The tuple sets are closed up to the diagonal, and they push forward naturally through factor maps (Li et al., 16 Aug 2025).

This localization program has a precursor in the local pair theory of Garcia-Ramos and Gutman. That paper does not define a numerical relative invariant for a factor map, but it introduces mean dimension pairs, proves that such pairs descend through factor maps when not collapsed, and relates them to the universal zero sofic mean dimension factor. In that sense it provides a factor-sensitive local language that later tuple theories refine into explicit relative invariants (García-Ramos et al., 2024).

5. Algebraic dynamics and exact identifications

The algebraic category supplies the most rigid relative theory currently available. Li and Liang introduced relative sofic mean length

$\rho$7

for $\rho$8-modules $\rho$9, together with relative sofic mean topological dimension, relative sofic metric mean dimension, relative topological entropy, and relative von Neumann–Lück rank. Their dynamical invariant $X$0 is defined from the cover complexity of the image model spaces $X$1, but now using an ultralimit $X$2 rather than an ordinary limsup. The introduction states that these relative invariants are completely a sofic phenomenon, since $X$3 does not depend on $X$4 when $X$5 is amenable and $X$6 is locally $X$7-finite (Li et al., 2015).

For countable $X$8-modules $X$9, the main algebraic identification is

$F\subset G$0

so relative sofic mean dimension and relative sofic metric mean dimension coincide with relative von Neumann–Lück rank on algebraic actions. In the same setting, a short exact sequence

$F\subset G$1

of compact metrizable abelian groups with $F\subset G$2-actions by automorphisms satisfies the dynamical addition formula

$F\subset G$3

These results make the algebraic theory the most explicit source of exact relative formulas (Li et al., 2015).

Hayes’s earlier work on algebraic actions already contained a restricted relative metric theory. For an inclusion $F\subset G$4 of $F\subset G$5-modules, he defines a relative $F\subset G$6-metric mean dimension by requiring ambient microstates on $F\subset G$7 to be small on a generating sequence $F\subset G$8. Proposition 4.3 then shows

$F\subset G$9

so in the algebraic quotient setting the relative invariant is exactly the quotient invariant. Combined with the theorem

$$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$00

for finitely generated $$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$01, this identifies quotient-relative geometry with von Neumann–Lück rank in a precise way (Hayes, 2013).

6. Amenable prototypes, metric refinements, and open problems

A useful amenable prototype is the 2025 paper on induced factors of amenable actions. For a countably infinite amenable group and a factor map $$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$02, the induced factor map on probability measures

$$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$03

satisfies

$$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$04

and

$$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$05

That paper does not discuss sofic groups, but it explicitly presents these results as highly relevant amenable-model analogues for questions one would ask about relative sofic mean dimension, especially the possibility that relative entropy in the original factor is converted, after induction to measures, into arbitrarily large simplex geometry and hence infinite relative mean dimension (Liu et al., 22 Nov 2025).

On the metric side, the relationship between $$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$06- and $$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$07-based formulations has been stabilized. The 2025 paper on sofic $$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$08-metric mean dimension proves

$$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$09

for $$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$10, and establishes a product formula with upper and lower bounds. The paper does not develop a relative theory, but its norm-comparison estimate and product-space inclusions are explicitly presented as adaptable to relative or conditional sofic mean dimension (Li, 4 Mar 2025).

Several issues remain open or unsettled. In Liang’s conditional theory, the equality

$$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$11

is posed as an open question at the topological level, even though the metric versions are equal (Liang, 2024). In the localization framework of $$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$12, the invariants are defined with respect to a fixed sofic approximation sequence $$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$13, and the paper does not claim independence from $$\pi^{d_i}(\operatorname{Map}(\cdot))\subseteq Y^{d_i}$$14 (Li et al., 16 Aug 2025). More broadly, the current literature retains at least two non-equivalent notions under the same heading “relative sofic mean dimension”: one fiberwise, one image-model-based. The field is therefore structurally rich but not yet terminologically unified (Liang, 2024, Li et al., 16 Aug 2025).