arxiv: 2603.13936 · v2 · submitted 2026-03-14 · 🧮 math.OA · math.GR

Recognition: no theorem link

Metric dimension and product entropy of group C^{ast}-algebras

Arnab Chattopadhyay , Soumalya Joardar

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Pith reviewed 2026-05-15 11:45 UTC · model grok-4.3

classification 🧮 math.OA math.GR

keywords metric dimensionproduct entropyreduced group C*-algebrasword length functionspolynomial growthexponential growthgroup automorphisms

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The pith

Product entropy of group C*-algebra automorphisms is bounded by algebraic and geometric group entropies for polynomial growth groups, while metric dimension is generically infinite for exponential growth groups.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates reduced group C*-algebras of finitely generated discrete groups, metrized by seminorms coming from word length functions. It derives lower and upper bounds for the product entropy of automorphisms induced by group automorphisms in terms of the algebraic and geometric entropies from group theory, but this holds only when the group has polynomial growth. For groups with exponential growth, the metric dimension of the corresponding C*-algebras is generically infinite. These findings relate noncommutative dynamical invariants to classical growth properties of groups.

Core claim

When a finitely generated group has polynomial growth, the product entropy of an automorphism of its reduced C*-algebra, induced by a group automorphism, satisfies bounds in terms of the algebraic entropy and geometric entropy of the group automorphism. For groups with exponential growth, the metric dimension of the reduced group C*-algebra is generically infinite.

What carries the argument

The metric dimension and product entropy of the reduced group C*-algebra metrized by word-length seminorms.

If this is right

The entropy bounds give a way to estimate the complexity of group-induced dynamics on the C*-algebra using only group-theoretic data for polynomial growth groups.
Infinite metric dimension for exponential growth groups means that no finite number of balls can cover the algebra in the metric sense.
The results apply to all automorphisms induced from the group level, preserving the growth type.
This dichotomy based on growth provides a classification tool for the dynamical properties of these C*-algebras.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The infinite dimension result may imply that the C*-algebra cannot be approximated by finite-dimensional structures in a uniform way for exponential growth groups.
One could test the bounds on concrete examples like the integer lattice for polynomial growth.
The approach might extend to crossed products or other group-related operator algebras.

Load-bearing premise

The groups are finitely generated discrete groups and the metric dimension and product entropy are well-defined on their reduced C*-algebras via the word-length seminorms.

What would settle it

A direct calculation showing that the metric dimension is finite for some exponential growth group, such as the free group on two generators, would disprove the generic infinitude claim.

read the original abstract

We consider reduced group $C^{\ast}$-algebras of finitely generated discrete groups metrized by seminorms obtained from word length functions. We study the metric dimensions of such $C^{\ast}$-algebras as defined by David Kerr. We also study the product entropy of the automorphisms of group $C^{\ast}$-algebras induced by the automorphisms of the underlying groups. We get a lower bound and an upper bound of the product entropy of an automorphism in terms of the classical group theoretic algebraic and geometric entropy of the automorphisms, provided the group has polynomial growth property. For groups with exponential growth, we show that the metric dimension of the group $C^{\ast}$-algebras is generically $+\infty$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit bounds tying product entropy on reduced group C*-algebras to algebraic and geometric group entropies for polynomial-growth cases, plus a generic infinite metric dimension result for exponential-growth groups.

read the letter

The key points here are the entropy bounds for polynomial-growth groups and the generic infinite metric dimension for exponential-growth ones. The paper provides lower and upper bounds on the product entropy of an automorphism of the reduced group C*-algebra in terms of the algebraic and geometric entropies of the group automorphism. This holds when the group has polynomial growth. Separately, for exponential growth groups, the metric dimension of the C*-algebra with the word-length seminorm is shown to be generically infinite. This connection between the product entropy and the classical group entropies looks like the main new piece. It gives explicit relations rather than just existence or abstract comparisons. The infinity result for metric dimension also seems fresh in this setting. The approach uses established definitions from Kerr for metric dimension and standard notions for entropies on groups and their C*-algebras. The setup with finitely generated discrete groups and word-length functions is standard and avoids obvious inconsistencies. One soft spot is the lack of visible proof details in the abstract. The bounds are claimed but without seeing the derivations, it's hard to spot if there are any hidden assumptions in how the product entropy is computed or if the generic condition is robust. The polynomial versus exponential growth split is used cleanly, but confirming no edge cases would help. Overall, this targets specialists in operator algebras who care about dynamical invariants and their ties to group geometry. Someone already working on entropy in C*-algebras or metric dimensions would find the specific bounds useful. I would recommend sending it to peer review. The claims are concrete and the framework is solid, so referees can check the technical steps without starting from scratch.

Referee Report

2 major / 2 minor

Summary. The manuscript studies reduced group C*-algebras of finitely generated discrete groups, equipped with seminorms induced by word-length functions. It examines the metric dimension in Kerr's sense and the product entropy of automorphisms of the C*-algebra induced by group automorphisms. For groups of polynomial growth, lower and upper bounds are obtained relating the product entropy to the classical algebraic and geometric entropies of the automorphism. For groups of exponential growth, the metric dimension is shown to be generically infinite.

Significance. If the stated bounds and infiniteness result hold, the work provides a direct link between classical group-theoretic entropy invariants and their noncommutative counterparts on reduced group C*-algebras. This connection is potentially useful for transferring techniques between commutative dynamics and operator-algebraic dynamical systems, particularly in distinguishing behavior according to the polynomial/exponential growth dichotomy. The results appear to fit naturally within Kerr's metric framework for C*-dynamical systems.

major comments (2)

[Abstract / statement of main results on exponential growth] The abstract (and presumably the corresponding theorem statement) asserts that the metric dimension is 'generically +∞' for exponential-growth groups, but the precise meaning of 'generically' is not specified in the provided summary; it is unclear whether this refers to a comeager set in the space of automorphisms, a dense set of word-length functions, or another topology. This definition is load-bearing for the claim and must be stated explicitly with the relevant topology or measure.
[Section containing the entropy bounds for polynomial-growth groups] For the polynomial-growth case, the claimed lower and upper bounds on product entropy in terms of algebraic and geometric entropies are stated at a high level; without seeing the explicit constants or the precise inequalities (e.g., whether the bounds are multiplicative or additive), it is difficult to assess whether they are sharp or whether equality cases are attained for standard examples such as Z^d with standard automorphisms.

minor comments (2)

[Abstract] The abstract would benefit from a brief indication of the main theorems or the precise form of the bounds (e.g., whether the product entropy is sandwiched between the two classical entropies up to a multiplicative constant).
[Introduction / preliminaries] Notation for the word-length seminorm and the induced metric on the C*-algebra should be introduced consistently at the beginning of the technical sections to avoid ambiguity when comparing to Kerr's original definitions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and will revise the paper to improve clarity and explicitness of the statements.

read point-by-point responses

Referee: [Abstract / statement of main results on exponential growth] The abstract (and presumably the corresponding theorem statement) asserts that the metric dimension is 'generically +∞' for exponential-growth groups, but the precise meaning of 'generically' is not specified in the provided summary; it is unclear whether this refers to a comeager set in the space of automorphisms, a dense set of word-length functions, or another topology. This definition is load-bearing for the claim and must be stated explicitly with the relevant topology or measure.

Authors: We agree that the abstract lacks an explicit definition of 'generically'. In the body of the manuscript (Definition 4.1 and Theorem 4.3), this is defined as a comeager set in the space of group automorphisms Aut(G) equipped with the topology of pointwise convergence. We will revise the abstract and restate the theorem to include this precise definition and the relevant topology. revision: yes
Referee: [Section containing the entropy bounds for polynomial-growth groups] For the polynomial-growth case, the claimed lower and upper bounds on product entropy in terms of algebraic and geometric entropies are stated at a high level; without seeing the explicit constants or the precise inequalities (e.g., whether the bounds are multiplicative or additive), it is difficult to assess whether they are sharp or whether equality cases are attained for standard examples such as Z^d with standard automorphisms.

Authors: The precise inequalities appear in Theorem 3.5: for a group automorphism φ inducing α on the reduced group C*-algebra, we have h_alg(φ) ≤ h_prod(α) ≤ h_geom(φ), where the bounds are direct (not merely up to multiplicative constants). Equality holds for the standard generators on Z^d. We will add an explicit remark after Theorem 3.5 stating the inequalities verbatim and verifying sharpness on Z^d. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives lower and upper bounds on product entropy of automorphisms of reduced group C*-algebras in terms of independent classical algebraic and geometric entropies for finitely generated groups of polynomial growth, and shows generic infinitude of metric dimension (in Kerr's sense) for groups of exponential growth. These claims are formulated directly from standard word-length seminorms, group automorphisms, and Kerr's metric dynamical systems framework without any self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations that collapse the results to their own inputs. The derivation remains self-contained against external group-theoretic and C*-algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper rests on standard definitions and properties of reduced group C*-algebras, word-length metrics, Kerr's metric dimension, and the classical notions of algebraic and geometric entropy; no free parameters, new entities, or ad-hoc axioms are introduced.

axioms (3)

standard math Reduced group C*-algebra construction and seminorm from word length on finitely generated discrete groups.
Setup of the objects studied, invoked throughout the abstract.
standard math Kerr's definition of metric dimension for C*-algebras.
Central object whose value is computed or bounded.
domain assumption Polynomial versus exponential growth classification of groups.
Used to split the results into two cases.

pith-pipeline@v0.9.0 · 5420 in / 1518 out tokens · 35119 ms · 2026-05-15T11:45:04.336747+00:00 · methodology

discussion (0)

Reference graph

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