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arxiv: 2605.17814 · v1 · pith:UFMJ6MESnew · submitted 2026-05-18 · 🧮 math.GR · math.MG

Geometric properties of the golden ration Thompson's group

Denys Svetelik This is my paper

Pith reviewed 2026-05-20 01:16 UTC · model grok-4.3

classification 🧮 math.GR math.MG
keywords golden ratio Thompson groupsasynchronous rational groupGromov hyperbolicityhorofunction boundaryCayley graphmonoid presentationtopological full groupV_tau
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The pith

The golden ratio Thompson groups F_τ, T_τ and V_τ embed into the asynchronous rational group, and the Cayley graph of the monoid M = ⟨L, R : LR² = RL²⟩ is Gromov hyperbolic with a Cantor-like horofunction boundary.

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

The paper shows that the golden ratio Thompson groups F_τ, T_τ and V_τ all embed into the asynchronous rational group. It introduces the monoid M generated by L and R subject to the single relation LR² = RL², whose topological full group is identified with V_τ. A distance function is derived for the Cayley graph of M. The graph is proved Gromov hyperbolic, and its horofunction boundary is shown to be homeomorphic to a Cantor-like set that includes additional isolated points between each pair of breakpoints. These results connect the algebraic structure of the monoid directly to geometric features of the group V_τ.

Core claim

The authors establish that F_τ, T_τ and V_τ embed in the asynchronous rational group. For the monoid M = ⟨L, R : LR² = RL²⟩, whose topological full group is V_τ, they compute the distance function on its Cayley graph, prove that the graph is Gromov hyperbolic, and show that the horofunction boundary is homeomorphic to a space resembling a Cantor set with isolated points situated between each pair of breakpoints.

What carries the argument

The monoid M = ⟨L, R : LR² = RL²⟩, whose topological full group equals V_τ and whose Cayley graph carries the distance function, hyperbolicity, and horofunction boundary used to obtain the geometric conclusions.

If this is right

  • The embeddings allow geometric and algorithmic properties of the asynchronous rational group to be pulled back to F_τ, T_τ and V_τ.
  • Explicit distance formulas become available for words in the monoid M and therefore for elements of V_τ.
  • Gromov hyperbolicity implies that geodesic triangles in the monoid graph are uniformly thin, yielding negative curvature behavior.
  • The described horofunction boundary supplies a concrete topological model for the asymptotic directions of V_τ.

Where Pith is reading between the lines

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

  • The same monoid presentation technique could be applied to other parameterized Thompson groups to test whether their associated graphs are also hyperbolic.
  • The isolated points between Cantor-set breakpoints may correspond to finite-order elements or particular stabilizers in the group action on the boundary.
  • Embedding into the asynchronous rational group may make word problems or conjugacy questions for these golden-ratio groups decidable via automata methods.

Load-bearing premise

The monoid M with generators L and R and the relation LR² = RL² has topological full group exactly equal to V_τ, so that geometric properties of the monoid graph transfer directly to the group.

What would settle it

An element of V_τ that lies outside the image of the monoid M under the topological full group map, or an explicit sequence of four points in the Cayley graph of M whose geodesics fail the thin-triangle condition for Gromov hyperbolicity.

Figures

Figures reproduced from arXiv: 2605.17814 by Denys Svetelik.

Figure 1
Figure 1. Figure 1: The generators x0 and x1 as tree diagrams. We consider two elements of Vτ to be the same if they agree everywhere but on a finite set of points. For a more detailed introduction to Fτ Vτ , we refer the reader to [9]. Definition 2.2. [7, Definition 1.1] For the group Fτ , we define two types of carets: • an x-type caret has a long left edge and a short right edge. It subdivides an interval [x, y] into [x, x… view at source ↗
Figure 2
Figure 2. Figure 2: The generators y0 and y1 as tree diagrams [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The generators c1 and c2 as tree diagrams. The labels on the leaves determine which domain intervals map to which range intervals [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The generators π0 and π1 as tree diagrams. The labels on the leaves determine which domain intervals map to which range intervals. 3. Irrational slope Thompson groups as an asynchronous automaton In this section, we use automaton theory to explore the dynamical properties of Thomp￾son groups Fτ , Tτ and Vτ . In particular, we prove that the Thompson groups Fτ , Tτ and Vτ are isomorphic to a certain group o… view at source ↗
Figure 5
Figure 5. Figure 5: Automata that generate the Thompson group Vτ . We endow the set X with the discrete topology, and Xω with the product topology. This space is homeomorphic to the Cantor set, which implies that its topological type does not depend on the choice of X. For any finite word v ∈ X∗ , the set c(v) = {vu : u ∈ Xω} is open and closed in this topology. The family {c(v) : v ∈ X∗} forms a basis for the topology in Xω … view at source ↗
Figure 6
Figure 6. Figure 6: Carets of the y1 generator addressed by pairs L’s,R’s and 0’s, 1’s splitting the interval into subintervals of length τ and τ 2 and then further dividing the subinterval of length τ into sequential subintervals of length τ 2 and τ 3 . Or we can start by dividing the interval into subintervals of length τ 2 and τ . Remark 3.17. Recall that the tree diagrams for the generators xn and yn are read following a … view at source ↗
Figure 7
Figure 7. Figure 7: The X0 generator We distinguish 3 cases where the input word starts with 00, 01, and 1, corresponding to all possible inputs. x0q2q1(00ω) = x0q2(0000ω2) = x0fL4 q2(ω2) = fL2 q2(ω2) = q2(00ω2) = q2q1(0ω) = q2q1X0(00ω); x0q2q1(01ω) = x0q2(001ω2) = x0fL2Rq2(ω2) = fRL2 q2(ω2) = q2(100ω2) = q2q1(10ω) = q2q1X0(01ω); x0q2q1(1ω) = x0q2(1ω2) = x0fRq2(ω2) = fR2 q2(ω2) = q2(11ω2) = q2q1(11ω) = q2q1X0(1ω). Hence, q in… view at source ↗
Figure 8
Figure 8. Figure 8: A diagram representing the homomorphism of ϕ from G to Vτ . □ Since by Theorems 2.4 and 2.5 the generators of both Tτ and Fτ are contained in Vτ , we obtained the following two corollaries. Corollary 3.21. The group of homeomorphisms of {0, 1} ω generated by X0, X1, Y0 and Y1 is isomorphic to Fτ . See [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Automata of the the inverse maps of β, γ and id of the Thomp￾son group Vτ Corollary 3.22. The group of homeomorphisms of {0, 1} ω generated by X0, X1, Y0, Y1, C1 and C2 is isomorphic to Tτ . See [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The composition of states of N verify this by considering the following eight possible scenarios: γ 2 (0ω) = γ(1ω) = 0ω; γ 2 (1ω) = γ(0ω) = 1ω; γβ(0ω) = γ(0)β(ω) = 1β(ω); γβ(1ω) = γ 2 (ω); βγ(0ω) = β(1ω) = 1γ(ω); βγ(1ω) = β(0ω) = 0β(ω); β 2 (0ω) = β(0)β(ω) = 0β 2 (ω); β 2 (1ω) = β(1)γ(ω) = 1γ 2 (ω). See [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The Cayley graph Cay(M) of the monoid M with some labeled elements L, R, L −1 or R−1 with inverses indicating moving “backwards” along an edge L or R, respectively. The cone of an element x ∈ M, denoted by Cone(x), is the set of all elements in M that can be obtained by multiplying x on the right by any element of the monoid M, formally, Cone(x) = {x · m : m ∈ M}, in other words, m ∈ M is in the cone of x… view at source ↗
Figure 12
Figure 12. Figure 12: Cone(L) of the graph Cay(M) Proposition 4.7. Let n, m ≥ 0, then the unique geodesic code of π(L n , Rm) is L −nRm. See [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The graph Cay(M) with the L −nRm code and Cone(LR2 ) highlighted Note that (a + 2) + (m − 1) ≤ n + m − 1 = k − 1. Then, according to the induction hypothesis, the length of a code c(RL2+a , Rm) is a + m + 1. The code c(L −2−aRm−1 ) is of such length, this implies that a code c(L n , Rm) visits LR2 . Given that a code c(L n , Rm) visits LR2 , we observe that it can be decomposed as: c(L n , Rm) = c(L n , L… view at source ↗
Figure 14
Figure 14. Figure 14: The areas A1 - A5 on the Cayley graph M Before proving Proposition 4.9 we invite the reader to view [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The Cayley graph Cay(M) with labeled elements of IM Assume that the statement holds for all x such that |x| ≤ n. Now, let |x| = n + 1. We point out the key property of IM: [0, 1]L ∩ [0, 1]R = [0, 1]LR2 = [0, 1]RL2 . Without loss of generality, let x and y have no common prefix, let x start with R, and y start with L. Then [0, 1]Rx′ ⊆ [0, 1]Ly′ for some x ′ , y′ . This implies [0, 1]Rx′ ⊆ [0, 1]L, which is… view at source ↗
Figure 16
Figure 16. Figure 16: A directed multigraph Γ and its corresponding path language tree L(Γ, r) This theorem plays a key role in Section 5. Not only does it provide the necessary tools to find the horofunction boundary using a combinatorial approach, but it also allows us to visualize the horofunction boundary. A finite directed multigraph is a graph Γ = (V, E), satisfying the conditions of a finite directed graph with the only… view at source ↗
Figure 17
Figure 17. Figure 17: A hyperedge in red splitting Cone(L) and M\Cone(L) Definition 5.11. A hyperedge is a line that splits a space into 2 disjoint half-planes. In the context of the graph M, where the space is the set of vertices M, a hyperedge selects a specific subset of vertices. To visualize this concept, see [PITH_FULL_IMAGE:figures/full_fig_p036_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: A vector field pointing towards the half-plane, whose elements are closer to R than to 1 where x = mx′ . Now by Theorem 4.4 d(x ′ , L) < d(x ′ , LR). An analogous argument is applied to prove d(x ′ , R) < d(x ′ , RL). Now let d(x, mLR) < d(x, mL), now assume that there exists x /∈ Cone(m) that satisfies this condition. Then there exists a geodesic from mL to x that passes through mR but by Proposition 4.9… view at source ↗
Figure 19
Figure 19. Figure 19: A hyperedge showing Cone(mLR) ∪ Cone(mRL) Proof. The proof requires studying two cases where m′ exists and not. When m′ ∈/ M, the conclusion follows trivially. Let’s look at the first case where m′ ∈ M. We apply Lemma 5.12 to Cone(m′ ), see [PITH_FULL_IMAGE:figures/full_fig_p038_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: A hyperedge picking out elements that satisfy the inequality d(x, m) < d(x, mR) Proposition 5.15. Let x, m ∈ M, then there exists n ∈ M such that mL = nRi , where i is as large as possible. There exists n ′ ∈ M such that nL = n ′Rj , where j = max{0, 1}. Then the following equivalent statements are valid: (1) d(x, mL) < d(x, m) ⇔ x ∈ [PITH_FULL_IMAGE:figures/full_fig_p039_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Visualization of statements 1 and 2 of Proposition 5.15 every element n /∈ Cone(m) and its offsprings. In other words, for every x ∈ Cone(m), the distance vectors {m, mL} and {m, mR} determine the distance vectors outside Cone(m). We end this section with a theorem that is a final consequence of these propositions. For any m ∈ M we call the set Cone(pLR) ∪ Cone(pRL) a triangle in M. We call the sets Cone(… view at source ↗
Figure 22
Figure 22. Figure 22: Partition of 1-level infinite atoms A1(M) this issue, we introduce a double blowup along elements of Z[τ ] ∩ (0, 1), resulting in the introduced modified equations above. Theorem 5.18. The horofunction boundary ∂hM of M is naturally homeomorphic to Dτ . The remainder of the section is dedicated to proving this theorem. To understand the structure of the horofunction boundary ∂hM, we will decompose infinit… view at source ↗
Figure 23
Figure 23. Figure 23: Decomposition of the atom a1 into a2, b2, and c2 in the second level A2(M) We translate this into terms of intervals c2 ⊆ [0, 1]LR = [τ 3+ , τ −], b2 ̸⊆ [0, 1]L2R2 = [0, τ 2− ], and b2 ̸⊆ [0, 1]LR2 = [τ 2+ , τ −]. Hence, c2 = {τ 2} is a point. □ Observe that the pairs of atoms a1, a2 and b1 and b2 have the same shape and type. Let c2 be an atom of type PL. Proposition 5.21. The infinite atom b1 at the 1-l… view at source ↗
Figure 24
Figure 24. Figure 24: Partition of 2-level infinite atoms A2(M) not decompose b1. We will refer to this infinite atom as d2. Note that b1 and d2 are atoms of different types. Let d2 be of type M2. From now on we will omit the formal proofs of atom decomposition, as essentially they follow the same format presented above. However, we invite the reader to convince themselves that the statements are valid. Corollary 5.22. The gra… view at source ↗
Figure 25
Figure 25. Figure 25: Decomposition of the atom d2 into f3, g3, and h3 in the se￾quential level A3(M) Proposition 5.23. The infinite atom d2 at 2-level decomposes into three disjoint infinite atoms in A3(M), denoted as f3, g3, and h3. These infinite atoms correspond to the following subsets of M: f3 = Cone(LR2L 2 ); g3 = Cone(L 2R 2 ) \ {Cone(LR2L 2 ) ∪ Cone(LR4 )}; h3 = Cone(LR4 ). Moreover, d2 = f3 ∪ g3 ∪ h3. See [PITH_FULL… view at source ↗
Figure 26
Figure 26. Figure 26: Representation of the decomposition of the atom f3 into j4, k4, and l4 the following subsets of M: j4 = Cone(LR2L) \ [PITH_FULL_IMAGE:figures/full_fig_p048_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: The type graph T of the infinite atoms in M with labeled edges [PITH_FULL_IMAGE:figures/full_fig_p049_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: The tree of atoms A(M) is generated by cylinder sets of the form Cα = {w ∈ P(T ) : w starts with α} for finite paths α. Proposition 5.26. The path space of the graph T depicted by [PITH_FULL_IMAGE:figures/full_fig_p049_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: The graphs Cay(M) and M′ Cay(M), we have 1 2 dCay(M)(x, y) ≤ dM′ [PITH_FULL_IMAGE:figures/full_fig_p054_29.png] view at source ↗
read the original abstract

We show that all three golden ratio Thompson's groups $F_\tau$, $T_\tau$ and $V_\tau$ embed in the asynchronous rational group. We prove properties of the Cayley graph of the monoid $M = \langle L, R : LR^2 = RL^2 \rangle$, whose topological full group is $V_\tau$. In particular, we compute a distance function for the Cayley graph of the monoid $M$. Additionally, we prove that this Cayley graph is hyperbolic in the sense of Gromov. Our analysis reveals that the horofunction boundary of this graph is homeomorphic to a space resembling a Cantor-like set, with additional isolated points situated between each pair of breakpoints.

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 / 2 minor

Summary. The paper claims that the golden ratio Thompson groups F_τ, T_τ and V_τ all embed into the asynchronous rational group. It studies the monoid M = ⟨L, R : LR² = RL²⟩, asserts that the topological full group of M is exactly V_τ, derives an explicit distance formula on the Cayley graph of M, proves that this graph is Gromov hyperbolic, and identifies the horofunction boundary with a Cantor-like set augmented by isolated points between breakpoints.

Significance. If the monoid-to-group identification is rigorously established and the geometric arguments are correct, the results would supply a concrete hyperbolic model for V_τ together with an explicit boundary description, strengthening the geometric toolkit available for Thompson groups and their embeddings. The distance formula and hyperbolicity proof would constitute verifiable, parameter-free contributions in the style of geometric group theory.

major comments (2)
  1. [Abstract and introduction] The central transfer of Gromov hyperbolicity and the horofunction boundary description from the Cayley graph of M to the group V_τ rests on the claim that the topological full group of M equals V_τ. No self-contained generation argument is supplied in the abstract linking the single relation LR² = RL² to the standard piecewise-linear generators of V_τ; this identification must be proved in detail (e.g., by exhibiting explicit words in L and R that realize the usual generators of V_τ and verifying that the action on the Cantor set coincides).
  2. [Section on embeddings] The embeddings of F_τ and T_τ into the asynchronous rational group are stated to follow from the same monoid framework. It is unclear whether these embeddings are proved independently of the V_τ identification or whether they inherit the same gap; a separate verification that the images of F_τ and T_τ lie inside the asynchronous rational group without invoking the full V_τ claim is needed.
minor comments (2)
  1. [Title] The title contains the typographical error 'ration' instead of 'ratio'.
  2. [Boundary section] Notation for the horofunction boundary should be introduced with a precise definition before the homeomorphism statement is asserted.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments, which will help improve the clarity of our manuscript. We address the major comments point by point below and plan to make revisions to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Abstract and introduction] The central transfer of Gromov hyperbolicity and the horofunction boundary description from the Cayley graph of M to the group V_τ rests on the claim that the topological full group of M equals V_τ. No self-contained generation argument is supplied in the abstract linking the single relation LR² = RL² to the standard piecewise-linear generators of V_τ; this identification must be proved in detail (e.g., by exhibiting explicit words in L and R that realize the usual generators of V_τ and verifying that the action on the Cantor set coincides).

    Authors: We acknowledge that the abstract is concise and does not include the full details of the identification. However, the manuscript provides the relation and asserts the topological full group is V_τ based on the action. To address this concern, we will revise the introduction to include a self-contained argument: we will exhibit explicit words in L and R corresponding to the standard generators of V_τ and verify the coincidence of the actions on the Cantor set. This will make the transfer of properties rigorous and self-contained. revision: yes

  2. Referee: [Section on embeddings] The embeddings of F_τ and T_τ into the asynchronous rational group are stated to follow from the same monoid framework. It is unclear whether these embeddings are proved independently of the V_τ identification or whether they inherit the same gap; a separate verification that the images of F_τ and T_τ lie inside the asynchronous rational group without invoking the full V_τ claim is needed.

    Authors: The embeddings for F_τ and T_τ are constructed directly using the monoid generators L and R without requiring the full identification with V_τ. Nevertheless, to clarify this, we will add a separate subsection verifying that the images of F_τ and T_τ are contained in the asynchronous rational group by explicit construction of the corresponding elements and their actions, independent of the V_τ claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rest on explicit monoid presentation and standard topological full group definitions.

full rationale

The paper explicitly defines the monoid M via the finite presentation ⟨L, R : LR² = RL²⟩ and derives its Cayley graph distance function, Gromov hyperbolicity, and horofunction boundary description directly from this presentation and the resulting word metric. The statement that the topological full group of M equals V_τ serves as a transfer mechanism to apply these results to the golden-ratio Thompson group, but this identification is presented as a known or separately established fact rather than a self-referential definition or fitted parameter within the geometric analysis. The embeddings of F_τ, T_τ, and V_τ into the asynchronous rational group are claimed as proven results without any exhibited reduction to prior fitted inputs or self-citation chains that would force the conclusions by construction. No equations or steps in the provided abstract or claims reduce the hyperbolicity or boundary homeomorphism to tautological inputs; the analysis remains self-contained against external benchmarks of group presentations and topological full groups.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard axioms of group presentations and the definition of topological full groups of monoids; no free parameters or new invented entities are visible in the abstract.

axioms (1)
  • domain assumption The monoid M is defined by the presentation ⟨L, R : LR² = RL²⟩ and its topological full group equals V_τ.
    This identification is required to link the geometric results on the Cayley graph of M to the group V_τ.

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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.Constants phi_golden_ratio echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    monoid M=⟨L,R:LR²=RL²⟩ whose topological full group is V_τ … horofunction boundary … homeomorphic to a space resembling a Cantor-like set, with additional isolated points situated between each pair of breakpoints … Z[τ]∩(0,1)

  • IndisputableMonolith.Foundation.BranchSelection branch_selection echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    every cone in Cay(M) is strongly geodesically convex … distance formula … self-similar to the whole graph itself

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.
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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

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