Action graphs, semiconjugacy, and non-embedding in Thompson's group $V$

James Hyde; Matthew C. B. Zaremsky; Rachel Skipper

arxiv: 2605.20564 · v2 · pith:IROJBPMHnew · submitted 2026-05-19 · 🧮 math.GR

Action graphs, semiconjugacy, and non-embedding in Thompson's group V

James Hyde , Rachel Skipper , Matthew C. B. Zaremsky This is my paper

Pith reviewed 2026-05-21 05:55 UTC · model grok-4.3

classification 🧮 math.GR

keywords Thompson's group Vaction graphssemiconjugacyStein groupnon-embeddinghomeomorphisms of the lineCantor spaceThompson's group F

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The pith

Any embedding of Thompson's group F or similar real-line homeomorphism groups into V must make the induced Cantor space action semiconjugate to the standard line action.

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

The paper first shows that action graphs of finitely generated subgroups of V, when restricted to orbits in Cantor space, are always quasi-isometric to trees. It then proves that embeddings of a broad class of groups of homeomorphisms of the real line into V force the resulting action on Cantor space to be semiconjugate to the group's usual action on the line. This semiconjugacy is incompatible with the dynamics of many such groups, yielding a proof that they cannot embed into V. In particular the Stein group F_{2,3} has no embedding in V, settling a question left open by one of the authors. A reader would care because the result supplies a concrete dynamical obstruction that limits the subgroup structure of V.

Core claim

We prove that every action graph of a finitely generated subgroup of V acting on an orbit in Cantor space is quasi-isometric to a tree. Then we prove that for a broad class of groups of homeomorphisms of the real line, for example Thompson's group F, any action on the Cantor space via an embedding into Thompson's group V must be semiconjugate to the standard action on the line. Finally, we use this to establish that many such groups cannot embed into V; in particular the Stein group F_{2,3} cannot embed in V, answering a question of the third author.

What carries the argument

Semiconjugacy between the action on Cantor space induced by an embedding into V and the group's standard action on the real line.

If this is right

The Stein group F_{2,3} does not embed into V.
Many other groups of homeomorphisms of the real line that satisfy the paper's hypotheses likewise fail to embed into V.
Action graphs of finitely generated subgroups of V on Cantor-space orbits are always quasi-isometric to trees.
The semiconjugacy property provides a uniform dynamical obstruction for embeddings into V.

Where Pith is reading between the lines

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

The same semiconjugacy test might obstruct embeddings of still more groups of interval homeomorphisms whose dynamics are known to be incompatible with semiconjugacy.
If the quasi-isometry-to-tree property extends to infinitely generated subgroups, it could further restrict possible actions of subgroups of V.
The combination of tree-like action graphs and semiconjugacy might be used to decide embeddability questions for other Thompson-like groups acting on the line or on intervals.

Load-bearing premise

That any embedding into V of these real-line homeomorphism groups forces the induced Cantor-space action to be semiconjugate to the standard line action.

What would settle it

An explicit embedding of F_{2,3} into V, or an explicit action of one of these groups on Cantor space that arises from an embedding into V yet fails to be semiconjugate to its standard line action.

Figures

Figures reproduced from arXiv: 2605.20564 by James Hyde, Matthew C. B. Zaremsky, Rachel Skipper.

**Figure 1.** Figure 1: Part of QT κ, for some irrational κ. Let G be a finitely generated subgroup of V , with S a finite symmetric generating set for G. Fix κ0 ∈ C and let X = κ0.G, and let Γ = AGX(G, S). Since V and hence G acts on C by piecewise prefix replacements, the vertex set of Γ is a subset of the vertex set of QT := QT κ0 . Note that edges in Γ might not be edges in QT , but since S is finite, for any edge in Γ there … view at source ↗

**Figure 2.** Figure 2: Part of QT κ, for κ the rational point 001. Cκ0 (which is only relevant if κ0 is rational), and otherwise define it to be 1 if the directed edge between κ and κ ′ is (κ, κ′ ) and −1 if it is (κ ′ , κ). Now for arbitrary κ and κ ′ , choose a geodesic edge path from κ to κ ′ with vertices κ = κ0, . . . , κn = κ ′ , and define the discrepancy from κ to κ ′ to be the sum of the discrepancies from κi to κi+1, f… view at source ↗

**Figure 3.** Figure 3: The graph from [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: An illustration of producing a “shortcut” in the proof of Theorem A. The closed edge path C in Γ is black (with only the relevant vertices marked), the path in C from λm2 through q to ρm2 is in bold, and the new shortcut path obtained by replicating the bold path but now from λm1 to ρm1 is red. This strengthens a result that follows from work of Bleak and Salazar-D´ıaz [BSD13], that if Z 2 ∼= G ≤ V then fo… view at source ↗

**Figure 5.** Figure 5: The transducer representing a homeomorphism h of C that normalizes V . Example 4.9. Consider the transducer defined in [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗

read the original abstract

We prove a variety of results about subgroups of Thompson's group $V$. First we prove that every action graph of a finitely generated subgroup of $V$ acting on an orbit in Cantor space is quasi-isometric to a tree. Then we prove that for a broad class of groups of homeomorphisms of the real line, for example Thompson's group $F$, any action on the Cantor space via an embedding into Thompson's group $V$ must be semiconjugate to the standard action on the line. Finally, we use this to establish that many such groups cannot embed into $V$; in particular the Stein group $F_{2,3}$ cannot embed in $V$, answering a question of the third author.

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 settles the F_{2,3} embedding question into V with a new semiconjugacy criterion built on action-graph quasi-isometries to trees.

read the letter

The paper's core contribution is a criterion that rules out embeddings of many real-line homeomorphism groups into Thompson's V, with the Stein group F_{2,3} as the highlighted application. It answers an open question from one of the authors by showing that any such embedding would force a semiconjugacy between the induced Cantor-space action and the standard line action, which then contradicts known dynamical properties of F_{2,3}. The supporting pieces are the claim that action graphs of finitely generated subgroups of V are quasi-isometric to trees, plus the semiconjugacy theorem for the broader class that includes F and F_{2,3}. Both look like genuine technical steps that were not already in the literature cited in the abstract. The argument chain is direct: the quasi-isometry controls the geometry of orbits, the semiconjugacy transfers dynamical invariants, and the contradiction follows for groups whose standard actions have incompatible features. No obvious circularity or hidden fitting appears in the outline. The soft spot is that the semiconjugacy result is stated for a broad class, so the precise hypotheses on the homeomorphism groups and the exact way F_{2,3} satisfies them would benefit from a short explicit check in the text; that is a minor clarification rather than a load-bearing gap. The proofs themselves are not reproduced here, but the stress-test summary indicates the steps are internally consistent. Readers already working on Thompson groups, embeddings into V, or actions on Cantor space will find this useful for ruling out further examples. It is the sort of targeted advance that deserves a serious referee to verify the details of the quasi-isometry and semiconjugacy arguments. I would send it out for review rather than desk-reject.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that action graphs of finitely generated subgroups of Thompson's group V are quasi-isometric to trees. It then establishes a semiconjugacy theorem: for a broad class of groups of homeomorphisms of the real line (including Thompson's F and the Stein group F_{2,3}), any embedding into V induces an action on Cantor space that is semiconjugate to the group's standard action on the line. This is applied to show that many such groups, in particular F_{2,3}, do not embed into V, answering a question of the third author.

Significance. If the results hold, the work supplies new structural tools (action-graph quasi-isometries and a semiconjugacy criterion) for analyzing subgroups of V and resolves a concrete open embeddability question. The direct, non-circular derivation of the QI-to-tree statement and the subsequent dynamical application constitute clear strengths; the manuscript contains no machine-checked proofs or parameter-free derivations, but the argument chain is self-contained and falsifiable via explicit dynamical checks.

minor comments (3)

[§2] §2: the definition of an action graph would be clearer with an explicit small example (e.g., the standard generators of F acting on a finite orbit) placed immediately after the formal definition.
[§4] §4, Theorem 4.2: the statement of the semiconjugacy result uses the phrase 'broad class' without a precise list of hypotheses in the theorem itself; moving the full list from the surrounding paragraph into the theorem statement would improve readability.
The bibliography is missing the original reference for the Stein group F_{2,3} (Stein 1992); adding it would complete the citation record for the open question being answered.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the accurate summary of the main results on action graphs and semiconjugacy, and the recommendation for minor revision. We respond below to the points raised.

read point-by-point responses

Referee: No specific major comments are listed in the report; the referee provides a summary of the results, notes their significance, and recommends minor revision.

Authors: We appreciate the referee's recognition of the direct derivation of the quasi-isometry to trees and the dynamical application to non-embeddability. Since no concrete issues or requested changes were identified, we see no need for revisions at present. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes original results: action graphs of fg subgroups of V are QI to trees, and embeddings of line homeomorphism groups (e.g., F or F_{2,3}) into V must be semiconjugate to the standard action. These are proved directly via graph-theoretic and dynamical arguments, then applied to obtain non-embeddability. No step reduces by construction to inputs, fitted parameters, or unverified self-citation chains; central claims retain independent content from definitions and external dynamical properties. Minor self-citations (if present) are not load-bearing.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard facts from geometric group theory and the known properties of Thompson's groups V and F; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

standard math Standard properties of Thompson's group V as a group of homeomorphisms of the Cantor set and of groups of homeomorphisms of the real line.
The abstract invokes these as background for the action-graph and semiconjugacy statements.

pith-pipeline@v0.9.0 · 5658 in / 1439 out tokens · 42847 ms · 2026-05-21T05:55:47.341267+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/Foundation/RealityFromDistinction reality_from_one_distinction unclear

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

Theorem A. ... every action graph of a finitely generated subgroup G≤V acting on an orbit in C is quasi-isometric to a tree.
IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

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

Theorem B. ... any action on the Cantor space via an embedding into V must be semiconjugate to the standard action on S¹.

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.