Superharmonic functions on the Lamplighter graph of Thompson's group F

Maksym Chornyi

arxiv: 1907.01440 · v1 · pith:6TDTAYQUnew · submitted 2019-07-02 · 🧮 math.GR

Superharmonic functions on the Lamplighter graph of Thompson's group F

Maksym Chornyi This is my paper

Pith reviewed 2026-05-25 10:22 UTC · model grok-4.3

classification 🧮 math.GR

keywords amenabilitysuperharmonic functionslamplighter graphThompson's group Fgroup actionsrandom walksextensive amenability

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The pith

Extending an amenability test from groups to their actions on sets yields a criterion testable on a subclass of superharmonic functions for the lamplighter graph of Thompson's group F.

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

The paper extends a non-standard amenability test that uses random walks and superharmonic functions from the setting of groups to the setting of group actions on sets. It applies the extension to Thompson's group F by drawing on properties of extensive amenability, which reduces the general test to a subclass of superharmonic functions on the associated lamplighter graph. The work develops this criterion explicitly and checks it on the subclass, while leaving the amenability of F itself unresolved. A sympathetic reader would care because the criterion offers a concrete new handle on a long-open question about F.

Core claim

The paper shows that the non-standard amenability test based on random walks and superharmonic functions extends to group actions on sets. Invoking extensive amenability for the action of F reduces the general test to a subclass of superharmonic functions, which is then examined on the lamplighter graph of F. This produces a potentially useful criterion without settling whether F is amenable.

What carries the argument

The extension of the amenability test based on random walks and superharmonic functions to the setting of group actions on sets.

If this is right

The extended test supplies a concrete criterion that can be checked directly on the subclass of superharmonic functions for the lamplighter graph of F.
Properties of extensive amenability allow the reduction of the general test to this subclass.
The criterion applies to any group action possessing the same extensive amenability features used here.

Where Pith is reading between the lines

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

The subclass criterion could be checked by direct computation of the relevant superharmonic functions on finite approximations to the lamplighter graph.
The same reduction technique might apply to other groups whose actions satisfy extensive amenability but whose amenability status is unknown.
If the subclass satisfies the test while the full class does not, the gap would point to a specific obstruction outside the subclass.

Load-bearing premise

The properties of extensive amenability invoked for the action of F on the lamplighter graph suffice to reduce the general superharmonic-function test to the subclass examined.

What would settle it

A concrete superharmonic function on the lamplighter graph of F that lies outside the examined subclass and violates the inequality required by the amenability test.

Figures

Figures reproduced from arXiv: 1907.01440 by Maksym Chornyi.

**Figure 1.** Figure 1: Schreier graph of F acting on Z[ 1 2 ] Further on, we will only be interested in the general structure of the graph. The exact dyadic numbers corresponding to individual vertices are of little interest to us. We also note that the group Pf (X) ⋊ F is generated by a finite set {(∅, a),(∅, b),({p}, e)}. When no confusion occurs, we may abuse the notation and call them a, b and σ respectively. Roughly speakin… view at source ↗

**Figure 2.** Figure 2: Structure of subtrees Ti (with no hairs) 1/2 1/2 1/2 1/2 1/2 1/8 1/8 1 1 1 1 1/2 1/4 1/4 1/8 1/8 1/8 1/8 1/2 1/2 1/2 1/4 1/4 1/4 1/4 1/4 1/8 1/16 1/4 1/4 1/4 1/4 1/16 φ0 φ1 φ2 [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

**Figure 3.** Figure 3: Values of ϕ0, ϕ1, ϕ2 on Xsk around p The amenability criterion from 8 can be reformulated as follows: there is a sequence {En} such that for any word of length ≤ n representing an element g we have that | f(gEn) f(En) − 1| < 1 2n . (In previous sections we took the inequality | f(gEn) f(En) − 1| < 1 n instead, but 1 2n is more suited for the purposes of this construction). Proposition 5. Let En be a sequen… view at source ↗

**Figure 4.** Figure 4: Golden path for n = 2 take g ∈ {a, b} ∗ which pushes the nearest point of the path to bai .p. Note that ϕi is equal to 1 2 i on all points of the golden path except bai .p where it is 1 2 i+1 . In this way, if En contains no points in Ti , we have that f(En) = 1 2 i , however, with our construction of g, f(gEn) ≤ 1 2 i+1 . Also, if g ∈ {a, b} ∗ or g ∈ {a, b} ∗σ, as in our case, then fk(gEn) ≤ fk(En). This … view at source ↗

**Figure 5.** Figure 5: E2 (in grey) as a subset of X 1/16 1/32 1/32 φ3=1/8 φ3=1/8 φ3=1/8 [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗

**Figure 6.** Figure 6: Location of E3 (in grey) and points in X where ϕ3 < 1 8 9. Generalized min-functions Denote by T the subspace of sequences in (0, 1]N whose all but finitely many coordinates are equal to 1. Let g : T → R + be a function satisfying the following: • g is non-negative and non-decreasing in any variable; • g is concave, i.e. λg(u) + (1 − λ)g(v) ≤ g(λu + (1 − λ)v) for 0 ≤ λ ≤ 1 and u, v ∈ T; • g is symmetric, … view at source ↗

**Figure 7.** Figure 7: Fragment of graph Z with values of ϕ Proof. Assume such a sequence exists. As before, we can assume | f(Eng) f(En) − 1| < 1 n Consider two cases: Case 1. ∃N ∈ N ∀n ≥ N : f(En) = 1. This means that starting from some moment all En are subsets of the tail. Take g = pb, where p is the generator corresponding to the switch at e and b multiplies all elements by b (here we are assuming the right notation, i.e. w… view at source ↗

read the original abstract

The goal is to extend a non-standard amenability test for groups, based on random walks and superharmonic functions, to group actions on sets, and to apply it to Thompson's group F using certain properties of extensive amenability. While no conclusive answer regarding the amenability of F is given, the approach is helpful in developing a new potentially useful criterion and testing it on a significant subclass of superharmonic functions.

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

Extends an amenability test to actions and applies it to a subclass on the lamplighter graph of F, but reaches no conclusion on amenability itself.

read the letter

The main takeaway is that this paper takes an existing amenability test based on random walks and superharmonic functions, extends it from groups to group actions on sets, and then applies the extended version to Thompson's group F via the lamplighter graph. It stops short of any decision on whether F is amenable, which is the long-standing open question it references. The extension itself is the concrete new piece, and the authors use properties of extensive amenability to reduce the general test to a significant subclass of functions. That reduction step is presented clearly, and the paper is straightforward that the application does not settle the amenability issue. The honesty about scope is a plus, and the choice of the lamplighter graph gives the work a specific, checkable target. The soft spot is that the usefulness of the new criterion is not demonstrated beyond the subclass. The reduction depends on the extensive amenability properties holding in a way that makes the subclass representative, but without independent evidence that this moves the needle on F or on other groups, the claim of a potentially useful criterion rests on future strengthening. The math is formal rather than computational, so there are no data or parameter issues to flag, but the independence from prior work on extensive amenability needs verification in the body to avoid any overlap. This is aimed at specialists in geometric group theory who track amenability criteria or work with superharmonic functions on Cayley graphs. Someone already following the literature on Thompson's F or random-walk tests might find the extension worth trying on other examples. I would send it to peer review because the technical move is legitimate and the subfield can judge whether the reduction is solid and whether the subclass carries enough weight.

Referee Report

0 major / 1 minor

Summary. The paper extends a non-standard amenability test for groups (based on random walks and superharmonic functions) to the setting of group actions on sets. It then applies the resulting criterion to the action of Thompson's group F on the lamplighter graph, invoking properties of extensive amenability to reduce the general test to a significant subclass of superharmonic functions. No conclusive determination is reached regarding the amenability of F; the central claim is that the extended criterion is potentially useful and has been successfully tested on the indicated subclass.

Significance. If the extension of the criterion is correctly formulated and the reduction to the examined subclass is valid, the work supplies a new technical tool for amenability questions involving group actions with complicated structure, such as those arising for Thompson's group F. The explicit verification on a concrete subclass constitutes a tangible demonstration of applicability rather than a purely formal extension, which strengthens the contribution in an area where amenability remains open.

minor comments (1)

The abstract and introduction would benefit from a brief explicit statement of the precise form of the extended criterion (e.g., the functional inequality or the random-walk condition that must be checked) before describing its application to F.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. No specific major comments were raised in the report, so we have no point-by-point revisions to address. The manuscript stands as submitted, with the extension of the superharmonic amenability criterion to actions and its application to the lamplighter graph of F presented as a technical tool without resolving the amenability question.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained against external benchmarks

full rationale

The provided abstract and reader's summary describe an extension of an existing amenability test (based on random walks and superharmonic functions) to group actions, followed by application to a subclass for Thompson's group F via properties of extensive amenability. No equations, definitions, or claims in the visible text reduce a prediction or central result to a fitted input, self-definition, or self-citation chain by construction. The paper explicitly refrains from resolving amenability of F, and the cited properties are treated as external enabling assumptions rather than derived within the work. This matches the default case of a self-contained derivation with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the work rests on standard definitions of amenability, superharmonic functions, and extensive amenability from prior literature.

pith-pipeline@v0.9.0 · 5580 in / 1107 out tokens · 30478 ms · 2026-05-25T10:22:06.593006+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 8. The action of G on X is amenable if and only if for all positive superharmonic functions f ... there exists a sequence {En} ... f(xEn)/f(En)→1
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We extend the statement to all amenable actions of a discrete group G on a graph X

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 4 internal anchors

[1]

Wolfgang Woess , Random Walks on Inﬁnite Graphs and Groups (Cam- bridge Tracts in Mathematics)

work page
[2]

Jos´ e Burillo , Introduction to Thompson ’s group F https://mat-web.upc.edu/people/pep.burillo/F%20book.pdf SUPERHARMONIC FUNCTIONS ON THE LAMPLIGHTER GRAPH 29

work page
[3]

Cannon, W.J

J.W. Cannon, W.J. Floyd, W.R. Parry , In- troductory notes on Richard Thompson ’s groups http://people.math.binghamton.edu/matt/thompson/cfp.pdf

work page
[4]

Daniel Yeow , Introduction to Thompson ’s group F (Honours The- sis) http://www.danielyeow.com/wp-content/uploads/2009/06/ honoursthesisfinal.pdf

work page 2009
[5]

Computational explorations in Thompson's group F

J. Burillo, S. Cleary, B. Wiest Computational explorations in Thomp- son ’s group Fhttps://arxiv.org/pdf/math/0506346.pdf

work page internal anchor Pith review Pith/arXiv arXiv
[6]

Azer Akhmedov , Non-amenability of R.Thompson ’s group F https://arxiv.org/abs/0902.3849

work page internal anchor Pith review Pith/arXiv arXiv
[7]

E.T.Shavgulidze, About amenability of subgroups of the group of diﬀeomor- phisms of the interval https://arxiv.org/abs/0906.0107

work page internal anchor Pith review Pith/arXiv arXiv
[8]

, Ex- tensive amenability and an application to interval exchang es

Juschenko, K., Matte Bon, N., Monod, N., de la Salle, M. , Ex- tensive amenability and an application to interval exchang es. arXiv preprint arXiv:1503.04977

work page arXiv
[9]

, Invariant means for the wobbling group

Juschenko, K., de la Salle, M. , Invariant means for the wobbling group

work page
[10]

, Amenability

Juschenko, K. , Amenability. Book in preparation. http://www.math.northwestern.edu/~juschenk/book.html

work page
[11]

Sam Northshield , Amenability and Superharmonic Functions https://digitalcommons.plattsburgh.edu/mathematics facpubs/19/

work page
[12]

Hartman, K

Y. Hartman, K. Juschenko, O. Tamuz, P. V. Ferdowsi . Thompson ’s group F is not strongly amenable https://arxiv.org/abs/1607.04915

work page arXiv
[13]

Some graphs related to Thompson's group F

Dmytro Savchuk . Some graphs related to Thompson ’s group F https://arxiv.org/abs/0803.0043

work page internal anchor Pith review Pith/arXiv arXiv

[1] [1]

Wolfgang Woess , Random Walks on Inﬁnite Graphs and Groups (Cam- bridge Tracts in Mathematics)

work page

[2] [2]

Jos´ e Burillo , Introduction to Thompson ’s group F https://mat-web.upc.edu/people/pep.burillo/F%20book.pdf SUPERHARMONIC FUNCTIONS ON THE LAMPLIGHTER GRAPH 29

work page

[3] [3]

Cannon, W.J

J.W. Cannon, W.J. Floyd, W.R. Parry , In- troductory notes on Richard Thompson ’s groups http://people.math.binghamton.edu/matt/thompson/cfp.pdf

work page

[4] [4]

Daniel Yeow , Introduction to Thompson ’s group F (Honours The- sis) http://www.danielyeow.com/wp-content/uploads/2009/06/ honoursthesisfinal.pdf

work page 2009

[5] [5]

Computational explorations in Thompson's group F

J. Burillo, S. Cleary, B. Wiest Computational explorations in Thomp- son ’s group Fhttps://arxiv.org/pdf/math/0506346.pdf

work page internal anchor Pith review Pith/arXiv arXiv

[6] [6]

Azer Akhmedov , Non-amenability of R.Thompson ’s group F https://arxiv.org/abs/0902.3849

work page internal anchor Pith review Pith/arXiv arXiv

[7] [7]

E.T.Shavgulidze, About amenability of subgroups of the group of diﬀeomor- phisms of the interval https://arxiv.org/abs/0906.0107

work page internal anchor Pith review Pith/arXiv arXiv

[8] [8]

, Ex- tensive amenability and an application to interval exchang es

Juschenko, K., Matte Bon, N., Monod, N., de la Salle, M. , Ex- tensive amenability and an application to interval exchang es. arXiv preprint arXiv:1503.04977

work page arXiv

[9] [9]

, Invariant means for the wobbling group

Juschenko, K., de la Salle, M. , Invariant means for the wobbling group

work page

[10] [10]

, Amenability

Juschenko, K. , Amenability. Book in preparation. http://www.math.northwestern.edu/~juschenk/book.html

work page

[11] [11]

Sam Northshield , Amenability and Superharmonic Functions https://digitalcommons.plattsburgh.edu/mathematics facpubs/19/

work page

[12] [12]

Hartman, K

Y. Hartman, K. Juschenko, O. Tamuz, P. V. Ferdowsi . Thompson ’s group F is not strongly amenable https://arxiv.org/abs/1607.04915

work page arXiv

[13] [13]

Some graphs related to Thompson's group F

Dmytro Savchuk . Some graphs related to Thompson ’s group F https://arxiv.org/abs/0803.0043

work page internal anchor Pith review Pith/arXiv arXiv