Another ambitable group

Jan Pachl

arxiv: 2606.06000 · v1 · pith:ZIUU4VDVnew · submitted 2026-06-04 · 🧮 math.GR · math.FA

Another ambitable group

Jan Pachl This is my paper

Pith reviewed 2026-06-27 23:24 UTC · model grok-4.3

classification 🧮 math.GR math.FA

keywords topological groupsambitableprecompacttopological group theory

0 comments

The pith

A topological group is ambitable without satisfying previously known sufficient conditions.

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

This note constructs a topological group that meets the definition of being ambitable. It does so without fulfilling any of the conditions that had previously been shown to imply ambitability, such as being precompact. The open problem asks if every topological group must be precompact or ambitable. This example demonstrates that ambitability can hold for reasons outside those conditions. Readers interested in topological groups would see this as expanding the known instances and testing the boundaries of the open question.

Core claim

The paper presents a topological group that is ambitable but does not satisfy previously known sufficient conditions for being so.

What carries the argument

The specific topological group constructed in the note, which acts as an example evading prior criteria while being ambitable.

If this is right

The class of ambitable topological groups includes examples not explained by earlier theorems.
The open question whether every topological group is precompact or ambitable receives an additional data point.
Classifications of when a topological group is ambitable must now include cases beyond the known sufficient conditions.

Where Pith is reading between the lines

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

This construction might be adaptable to produce a group that is neither precompact nor ambitable, settling the open question negatively.
It highlights the need for new sufficient conditions or a direct proof of the dichotomy.
Similar constructions could be explored in related areas like uniform spaces or semigroup actions.

Load-bearing premise

The constructed group is a valid topological group that is ambitable and does not meet any of the previously published sufficient conditions.

What would settle it

Verification that the presented group actually satisfies one of the known sufficient conditions or fails to be ambitable.

read the original abstract

It is an open question whether every topological group is precompact or ambitable. This note presents a topological group that is ambitable but does not satisfy previously known sufficient conditions for being so.

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

This short note claims one new ambitable topological group outside the listed sufficient conditions, but supplies no construction or checks in the abstract.

read the letter

This note is a brief announcement that there exists a topological group which is ambitable yet fails every previously published sufficient condition for that property.

What is new is the concrete example itself. It directly targets the open question of whether every topological group must be precompact or ambitable, and it positions the example as one that sits outside the known routes to ambitability. If the construction and the evasion checks hold, it adds a data point that separates the property from those conditions.

The paper does well by staying short and tying the claim straight to the open problem without extra framing. For readers already following this corner of topological group theory, the example could be worth recording once the details are confirmed.

The soft spot is the absence of any construction or explicit verification in the abstract. The central assertion requires showing that the object is a topological group, meets the definition of ambitable, and evades each prior sufficient condition. The stress-test note is right that this needs an exhaustive, cited list with line-by-line checks; without that, the claim cannot be assessed from what is given. Soundness is therefore provisional.

This is for specialists working on topological groups and related amenability-type properties. A reader tracking the specific open question might get value from the example if it survives scrutiny.

It deserves a serious referee because it claims a separating instance for an open problem, even though the note is minimal. I would recommend sending it to peer review in an appropriate specialized journal once the full text with the construction and checks is in hand.

Referee Report

1 major / 0 minor

Summary. The manuscript presents a topological group that is ambitable but does not satisfy previously known sufficient conditions for being ambitable, in the context of the open question whether every topological group is precompact or ambitable.

Significance. If the construction is valid and the claim that it evades all prior sufficient conditions is substantiated by explicit checks, the example would be significant for separating the ambitable property from the known sufficient conditions in the literature.

major comments (1)

[Abstract] Abstract: The central claim that the presented group 'does not satisfy previously known sufficient conditions for being so' requires an explicit, cited list of all such conditions together with a verification that none apply to the example. Without this enumeration and line-by-line check, the assertion that the example is new in this respect cannot be independently assessed and is load-bearing for the note's contribution.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for highlighting the need to make the novelty of the example fully verifiable. We address the single major comment below and will revise the manuscript to incorporate the requested clarification.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that the presented group 'does not satisfy previously known sufficient conditions for being so' requires an explicit, cited list of all such conditions together with a verification that none apply to the example. Without this enumeration and line-by-line check, the assertion that the example is new in this respect cannot be independently assessed and is load-bearing for the note's contribution.

Authors: We agree that the abstract's claim would be easier to assess with an explicit enumeration. In the revised version we will add a short section (or expanded introduction paragraph) that lists the known sufficient conditions from the literature on ambitable groups, with citations, followed by direct verification that none of them hold for the group constructed in the note. This addresses the load-bearing aspect of the contribution without altering the main construction or results. revision: yes

Circularity Check

0 steps flagged

No circularity: construction presented without self-referential derivation or load-bearing self-citation.

full rationale

The paper is a short note exhibiting a concrete topological group example claimed to be ambitable while evading prior sufficient conditions. No equations, fitted parameters, ansatzes, or derivations appear. The central claim rests on the explicit construction satisfying the definition of ambitable and failing listed prior conditions; this is an external verification task rather than an internal reduction to the paper's own inputs. No self-citation chains, uniqueness theorems, or renamings are invoked as load-bearing steps. The derivation chain is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5526 in / 830 out tokens · 17289 ms · 2026-06-27T23:24:53.752956+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references

[1]

College Publications, King’s College London, 2011

Kunen, K.Set theory. College Publications, King’s College London, 2011

2011
[2]

Topology Appl.156(13) (2009), 2200-2208

Pachl, J.Ambitable topological groups. Topology Appl.156(13) (2009), 2200-2208

2009
[3]

Corrections and supplements:http://www.fields.utoronto.ca/publications/supplements 3

Pachl, J.Uniform spaces and measures, Fields Institute Monographs, Springer, New York, 2013. Corrections and supplements:http://www.fields.utoronto.ca/publications/supplements 3

2013

[1] [1]

College Publications, King’s College London, 2011

Kunen, K.Set theory. College Publications, King’s College London, 2011

2011

[2] [2]

Topology Appl.156(13) (2009), 2200-2208

Pachl, J.Ambitable topological groups. Topology Appl.156(13) (2009), 2200-2208

2009

[3] [3]

Corrections and supplements:http://www.fields.utoronto.ca/publications/supplements 3

Pachl, J.Uniform spaces and measures, Fields Institute Monographs, Springer, New York, 2013. Corrections and supplements:http://www.fields.utoronto.ca/publications/supplements 3

2013