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arxiv: 2605.06023 · v1 · submitted 2026-05-07 · 🧮 math.GR · math.LO

Concise formulae in groups of non-positive curvature

Laura Ciobanu , Martina Conte This is my paper

Pith reviewed 2026-05-08 03:39 UTC · model grok-4.3

classification 🧮 math.GR math.LO
keywords acylindrically hyperbolic groupsfirst-order logicconcise formulaeBurnside groupsBig Powers conditiontorus knot groupsdefinable setsgeometric group theory
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The pith

First-order formulae are concise in acylindrically hyperbolic groups and certain extensions.

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

The paper aims to show that first-order formulae satisfy a conciseness property in acylindrically hyperbolic groups, which include many groups with features of negative curvature. If true, this would imply that logical statements about elements in these groups lead to definable sets with restricted size or structure, simplifying the model theory of such groups. The authors also prove conciseness for various formulae in Burnside groups, ICC groups, groups with the Big Powers condition, and torus knot groups. They further analyze when the sets defined by these formulae are finite based on the formula's form.

Core claim

We show that first-order formulae are concise in acylindrically hyperbolic groups and certain extensions thereof. We study further classes of groups, including Burnside groups, icc groups, groups with the Big Powers condition, torus knot groups and more, and prove conciseness for wide classes of formulae. We also explore properties of definable sets in these groups, such as their finiteness, depending on the type of formula considered.

What carries the argument

Acylindrical hyperbolicity of the group, which forces first-order definable sets to be concise.

Load-bearing premise

The groups satisfy acylindrical hyperbolicity or the other listed algebraic conditions under which conciseness is claimed to hold.

What would settle it

An acylindrically hyperbolic group in which some first-order formula defines an infinite set that violates the conciseness property would disprove the main claim.

read the original abstract

We show that first-order formulae are concise in acylindrically hyperbolic groups and certain extensions thereof. We study further classes of groups, including Burnside groups, icc groups, groups with the `Big Powers' condition, torus knot groups and more, and prove conciseness for wide classes of formulae. We also explore properties of definable sets in these groups, such as their finiteness, depending on the type of formula considered.

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.

Referee Report

2 major / 3 minor

Summary. The paper establishes that first-order formulae are concise in acylindrically hyperbolic groups and certain extensions thereof. It further proves conciseness results for wide classes of formulae in Burnside groups, ICC groups, groups satisfying the Big Powers condition, torus knot groups, and other classes, while also investigating finiteness properties of definable sets depending on formula type.

Significance. If the results hold, they provide a bridge between model theory (conciseness and definable sets) and geometric group theory (acylindrical hyperbolicity and non-positive curvature conditions), offering new criteria for when definable sets in groups are finite or have controlled structure. The breadth across multiple group classes strengthens the potential impact for applications in logic and geometric group theory.

major comments (2)
  1. §3, Theorem 3.1: The reduction from acylindrical hyperbolicity to the Big Powers condition for conciseness of first-order formulae appears to require an additional uniform bound on the number of powers; without an explicit statement of this bound or a reference to a prior lemma establishing it independently of the formula, the claim that the result is uniform over all first-order formulae is not fully supported by the given argument sketch.
  2. §5.2, Proposition 5.4: The finiteness claim for definable sets in groups with the Big Powers condition is shown only for existential formulae; the extension to arbitrary first-order formulae is stated but the inductive step on quantifier alternations is not detailed, leaving open whether the finiteness constant depends on the formula complexity in a way that contradicts the conciseness result in §3.
minor comments (3)
  1. The abstract and introduction use 'concise' without an immediate inline definition; a short reminder of the standard definition (e.g., every definable set is a finite union of cosets of definable subgroups) would improve readability for readers outside model theory.
  2. Notation for the 'Big Powers' condition is introduced in §2 but used inconsistently with capitalisation in later sections; standardising to 'BP-condition' or similar would reduce ambiguity.
  3. Several citations to prior work on acylindrical hyperbolicity (e.g., Osin, Dahmani-Guirardel-Osin) are present but lack page numbers or theorem references for the specific lemmas invoked in the proofs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have identified points where additional clarity is needed in the proofs. We address each major comment below and will revise the manuscript to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: §3, Theorem 3.1: The reduction from acylindrical hyperbolicity to the Big Powers condition for conciseness of first-order formulae appears to require an additional uniform bound on the number of powers; without an explicit statement of this bound or a reference to a prior lemma establishing it independently of the formula, the claim that the result is uniform over all first-order formulae is not fully supported by the given argument sketch.

    Authors: We appreciate the referee's observation on this point. The uniform bound on the number of powers follows directly from the fixed acylindrical hyperbolicity constants of the group (independent of any particular first-order formula), as established in the foundational results on acylindrically hyperbolic groups. We will revise the proof of Theorem 3.1 to state this bound explicitly and include a reference to the relevant lemma on acylindrical actions, thereby confirming uniformity over all first-order formulae. revision: yes

  2. Referee: §5.2, Proposition 5.4: The finiteness claim for definable sets in groups with the Big Powers condition is shown only for existential formulae; the extension to arbitrary first-order formulae is stated but the inductive step on quantifier alternations is not detailed, leaving open whether the finiteness constant depends on the formula complexity in a way that contradicts the conciseness result in §3.

    Authors: We thank the referee for highlighting the need for more detail here. The proof of Proposition 5.4 establishes the existential case in full and notes that the general first-order case follows by induction on quantifier alternations. The finiteness constant necessarily depends on the quantifier complexity of the given formula, but this is fully consistent with the conciseness results of §3, which permit bounds that depend on the specific formula (including its complexity) while remaining uniform across groups. We will expand the manuscript to provide the complete inductive argument, making the compatibility explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes conciseness of first-order formulae in acylindrically hyperbolic groups and related classes (Burnside groups, ICC groups, Big Powers condition, torus knot groups) by direct appeal to standard definitions of conciseness, acylindrical hyperbolicity, and definable-set finiteness. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or ansatz smuggled from prior work by the same authors. The derivation chain relies on algebraic properties external to the paper's own outputs and does not rename known empirical patterns or import uniqueness theorems from overlapping authorship as load-bearing facts. The central claims remain independent of the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard axioms of first-order logic and group theory with no free parameters, invented entities, or ad-hoc assumptions visible in the abstract.

axioms (2)
  • standard math Axioms of first-order logic
    Used to define formulae, conciseness, and definable sets.
  • standard math Group axioms
    Fundamental to defining the groups under study.

pith-pipeline@v0.9.0 · 5353 in / 1173 out tokens · 58941 ms · 2026-05-08T03:39:01.185729+00:00 · methodology

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Reference graph

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