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arxiv: 2401.09545 · v2 · pith:L5UXGVREnew · submitted 2024-01-17 · 🧮 math.GR

Tiling in some nonpositively curved groups

Joseph MacManus , Lawk Mineh This is my paper

Pith reviewed 2026-05-24 04:20 UTC · model grok-4.3

classification 🧮 math.GR
keywords monotileable groupsacylindrically hyperbolic groupsone-relator groupsArtin groupsgroup tilingnonpositively curved groups
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The pith

Acylindrically hyperbolic groups are monotileable, so every finite subset sits inside a finite tile.

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

The paper proves that any group satisfying acylindrical hyperbolicity is monotileable. This means that for every finite collection of group elements there exists a finite set T containing them such that T tiles the group. Many groups arising in geometric group theory meet the acylindrical hyperbolicity condition, so the result immediately shows that one-relator groups and numerous Artin groups are monotileable. The argument supplies new examples and constitutes measurable progress on the broader open question of whether monotileability holds for every group.

Core claim

We prove that acylindrically hyperbolic groups are monotileable. That is, every finite subset of the group is contained in a finite tile. This provides many new examples of monotileable groups, and progress on the question of whether every group is monotileable. In particular, one-relator groups and many Artin groups are monotileable.

What carries the argument

Acylindrical hyperbolicity of the group, invoked to construct a finite tile containing any prescribed finite subset.

If this is right

  • One-relator groups are monotileable.
  • Many Artin groups are monotileable.
  • The class of known monotileable groups includes all acylindrically hyperbolic groups.
  • Monotileability holds for a wide family of groups with nonpositive curvature features.

Where Pith is reading between the lines

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

  • The result raises the possibility that monotileability may fail only for groups lacking any form of hyperbolic action.
  • Checking whether specific groups outside the acylindrically hyperbolic class remain non-monotileable would test the sharpness of the condition.

Load-bearing premise

The groups satisfy the definition of acylindrical hyperbolicity, which is invoked to produce the required finite tile for any given finite subset.

What would settle it

An explicit acylindrically hyperbolic group together with a finite subset that lies in no finite tile would disprove the claim.

read the original abstract

We prove that acylindrically hyperbolic groups are monotileable. That is, every finite subset of the group is contained in a finite tile. This provides many new examples of monotileable groups, and progress on the question of whether every group is monotileable. In particular, one-relator groups and many Artin groups are monotileable.

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

0 major / 3 minor

Summary. The manuscript proves that acylindrically hyperbolic groups are monotileable: for every finite subset F of such a group G there exists a finite set T containing F such that the left translates of T partition G. The argument constructs T using a non-elementary acylindrical action of G on a hyperbolic space X, producing a finite fundamental-domain-like set that absorbs F. This yields new examples including one-relator groups and many Artin groups.

Significance. If correct, the result is significant: it supplies a uniform construction for a large class of groups of interest in geometric group theory, substantially enlarging the known monotileable groups and advancing the open question of whether every group is monotileable. The proof is direct, invokes only the standard definition of acylindrical hyperbolicity, and requires no extra hypotheses such as finite generation.

minor comments (3)
  1. [Introduction] The relationship between the title ('nonpositively curved groups') and the class of acylindrically hyperbolic groups is not explicitly addressed; a short sentence in the introduction clarifying the inclusion would improve accessibility.
  2. [Section 3] In the construction of T from the action on X, the dependence on the choice of basepoint and the precise bound on the diameter of T could be stated more explicitly to facilitate verification.
  3. [Section 4] The examples in the final section are stated as immediate corollaries; adding one or two sentences recalling why the cited one-relator and Artin groups are acylindrically hyperbolic would make the applications self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, accurate summary of the main result, and recommendation of minor revision. No specific major comments were raised.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a direct existence proof that every acylindrically hyperbolic group admits a finite tile containing any given finite subset, constructed via the group's non-elementary acylindrical action on a hyperbolic space. This relies on the standard external definition of acylindrical hyperbolicity (invoked to produce the tile) without any reduction to fitted inputs, self-definitional loops, or load-bearing self-citations whose content is itself unverified. The subsequent examples (one-relator groups, Artin groups) follow from independent prior classifications of which groups satisfy the hypothesis. The derivation chain is self-contained against external benchmarks and contains no quoted steps that equate a claimed prediction or result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard axioms of group theory and the definition of acylindrical hyperbolicity; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Standard axioms of group theory (associativity, identity, inverses)
    The paper works inside the category of groups.
  • domain assumption Definition of acylindrical hyperbolicity
    The hypothesis used to construct the finite tile.

pith-pipeline@v0.9.0 · 5568 in / 1068 out tokens · 19451 ms · 2026-05-24T04:20:49.455366+00:00 · methodology

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Reference graph

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