arxiv: 2604.24556 · v1 · submitted 2026-04-27 · 🧮 math.DS · math.GR

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Algebraic expansivity on abelian groups

Mauricio Achigar

Authors on Pith no claims yet

Pith reviewed 2026-05-07 17:34 UTC · model grok-4.3

classification 🧮 math.DS math.GR

keywords algebraic expansivityabelian groupsPontryagin dualitytopological expansivityalgebraic entropyexpansive mapsdynamical systems

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The pith

Algebraic expansivity on torsion abelian groups is exactly the dual of topological expansivity on totally disconnected compact groups under Pontryagin duality.

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

The paper introduces algebraic expansivity as an algebraic version of the separation property for endomorphisms of abelian groups. It develops the basic features of this notion, links it to algebraic entropy, and proves that positively expansive epimorphisms exist only on finite groups. The central result uses Pontryagin duality to show that algebraic expansivity on torsion groups corresponds exactly to topological expansivity on their dual totally disconnected compact groups. This creates a direct bridge between algebraic dynamics on discrete groups and topological dynamics on compact spaces.

Core claim

Algebraic expansivity on torsion abelian groups is exactly the dual property of topological expansivity on totally disconnected compact groups. The paper proves this equivalence via Pontryagin duality, shows that the algebraic property relates to Weiss's algebraic entropy, and establishes that positively expansive epimorphisms of abelian groups are possible only when the group is finite.

What carries the argument

Algebraic expansivity, defined for endomorphisms of abelian groups as the property that every nonzero element is separated from zero under some iterate, transferred exactly by Pontryagin duality to topological expansivity on the dual compact group.

If this is right

Positively expansive epimorphisms of abelian groups exist only when the group is finite.
Algebraic expansivity corresponds directly to Weiss's algebraic entropy in the same setting.
The duality transfers expansivity and related separation properties between the algebraic and topological categories.
Infinite abelian groups admit no positively expansive epimorphisms.

Where Pith is reading between the lines

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

The same duality technique could be applied to other dynamical properties such as entropy or mixing to obtain algebraic-topological correspondences.
Without the torsion assumption the duality would fail because the dual group would have a connected component.
The finiteness restriction suggests a classification problem: which algebraic actions on infinite groups can still exhibit weaker forms of expansivity.

Load-bearing premise

The groups are abelian, the endomorphisms meet the paper's definition of algebraic expansivity, and the groups are torsion so their duals are totally disconnected.

What would settle it

Exhibit a torsion abelian group together with an endomorphism that satisfies algebraic expansivity but whose Pontryagin dual map on the compact group fails to be topologically expansive.

read the original abstract

Building on the author's earlier work on topological and abstract expansivity, this paper introduces and explores the notion of algebraic expansivity for endomorphisms of abelian groups. We analyze the fundamental properties of this algebraic analogue, establish its relationship with Weiss's algebraic entropy, and prove that positively expansive epimorphisms are necessarily restricted to finite systems. Finally, we demonstrate a robust connection with topological dynamics via Pontryagin duality: algebraic expansivity on torsion abelian groups is shown to be exactly the dual property of topological expansivity on totally disconnected compact groups.

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 defines algebraic expansivity for endomorphisms of abelian groups and shows it is exactly dual to topological expansivity on the Pontryagin dual for torsion groups.

read the letter

This paper introduces algebraic expansivity for endomorphisms of abelian groups and proves that, for torsion groups, the property is exactly dual to topological expansivity on the totally disconnected compact dual group under Pontryagin duality. It also works out basic properties of the new notion, connects it to Weiss algebraic entropy, and shows that positively expansive epimorphisms must act on finite groups only. The duality equivalence is the clearest new statement. The definition is set up so the correspondence holds cleanly once torsion is assumed, and the arguments rely on standard facts about Pontryagin duality rather than new machinery. The finiteness result fits naturally into the development and does not appear forced. No internal contradictions or unsupported steps are visible from the claims and the stress-test note. The only soft spot worth noting is that the finiteness claim for expansive epimorphisms may overlap with earlier entropy bounds in the literature; the paper presents it as part of the package rather than a standalone advance. This is a specialized paper for readers already working in algebraic or topological dynamics on groups. Someone interested in transferring techniques between discrete and compact settings via duality would find the equivalence useful. It deserves peer review because the central duality statement is precise and checkable, even if the overall scope remains narrow.

Referee Report

0 major / 3 minor

Summary. The paper introduces algebraic expansivity as a new notion for endomorphisms of abelian groups. It studies basic properties of this concept, relates it to Weiss's algebraic entropy, proves that positively expansive epimorphisms are necessarily finite, and establishes a duality theorem showing that algebraic expansivity on torsion abelian groups is equivalent to topological expansivity on the Pontryagin duals (totally disconnected compact groups).

Significance. If the central duality and finiteness results hold, the work supplies a precise algebraic counterpart to topological expansivity that enables transfer of techniques between discrete group endomorphisms and compact topological dynamics. The explicit link to algebraic entropy and the restriction of expansive epimorphisms to finite systems are concrete contributions that could support further results in algebraic dynamics.

minor comments (3)

The definition of algebraic expansivity (introduced in the opening sections) should be restated verbatim in the statement of the duality theorem to make the equivalence immediately verifiable without back-referencing.
In the proof that positively expansive epimorphisms are finite, clarify whether the argument uses only the new definition or also invokes standard facts about algebraic entropy; a short remark on this dependence would improve readability.
The abstract refers to 'the author's earlier work on topological and abstract expansivity'; adding the precise citation in the abstract or the first paragraph of the introduction would help readers locate the foundational results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of the manuscript, recognition of its significance, and recommendation for minor revision. We are pleased that the central duality theorem linking algebraic expansivity on torsion abelian groups to topological expansivity on totally disconnected compact groups, along with the finiteness result for positively expansive epimorphisms and the connection to algebraic entropy, has been noted as a concrete contribution.

Circularity Check

0 steps flagged

Minor self-citation on context; core definition, entropy link, and duality equivalence are independently derived

full rationale

The paper defines algebraic expansivity anew for endomorphisms of abelian groups, proves its relation to Weiss algebraic entropy, establishes that positively expansive epimorphisms must be finite, and demonstrates via Pontryagin duality that the property on torsion groups is dual to topological expansivity on the compact duals. Citations to the author's prior topological expansivity work supply background definitions and motivation but do not serve as unverified load-bearing premises for the new equivalence or finiteness result; the duality proof relies on standard facts about Pontryagin duality for discrete torsion groups rather than reducing to a self-citation chain or by-construction renaming. No fitted parameters, self-definitional loops, or smuggled ansatzes appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper introduces one new defined concept and relies on one standard theorem from harmonic analysis; no free parameters or invented physical entities appear.

axioms (1)

standard math Pontryagin duality theorem for locally compact abelian groups
Invoked to equate algebraic expansivity on the discrete side with topological expansivity on the compact dual side.

invented entities (1)

algebraic expansivity no independent evidence
purpose: Algebraic analogue of expansivity for endomorphisms of abelian groups
New definition introduced to explore properties and duality; no independent falsifiable evidence outside the paper's own constructions.

pith-pipeline@v0.9.0 · 5365 in / 1186 out tokens · 81606 ms · 2026-05-07T17:34:59.565840+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 1 canonical work pages

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