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arxiv: 2605.06739 · v1 · submitted 2026-05-07 · 🧮 math.GR

Recognition: no theorem link

Forest Diagrams and Lengths for the Generalised Thompson's Group F(n)

Mart\'in G\'omez Reynolds
Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:20 UTC · model grok-4.3

classification 🧮 math.GR
keywords Thompson group F(n)forest diagramsword lengthdead end elementsn-ary forestsgeneralized Thompson groups
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The pith

Extended forest diagrams represent elements of F(n) as n-ary forests with leaf bijections and supply a new word-length formula.

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

The paper extends two-way forest diagrams to the generalized Thompson group F(n) by representing each element as a pair of infinite bounded n-ary forests together with an order-preserving bijection of their leaves. From these diagrams the authors extract an alternative expression for the word length of the element with respect to the standard generating set, different from the earlier Fordham-Cleary formula. Using the new length formula they re-establish that dead-end elements exist inside F(n) and prove that every such element has depth exactly two. A reader would care because the diagrams give a concrete, visual method for tracking lengths and special elements in these groups that arise in geometric group theory and dynamical systems.

Core claim

Every element of the generalized Thompson group F(n) can be represented by a pair of infinite, bounded n-ary forests together with an order-preserving bijection between their leaves, and the word length of the element equals a quantity read directly from the number and arrangement of carets in these diagrams.

What carries the argument

The extended two-way forest diagram: a pair of infinite bounded n-ary forests equipped with an order-preserving bijection of their leaves.

If this is right

  • A new explicit formula for word length in F(n) is obtained from the forest diagrams.
  • Dead-end elements exist inside F(n).
  • Every dead-end element of F(n) has depth exactly two.

Where Pith is reading between the lines

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

  • The diagrammatic length may be easier to compute by hand or machine than previous formulas for moderate n.
  • Similar forest representations could be tried for other Thompson-like groups or for subgroups of F(n).
  • The depth-two result suggests that the Cayley graph of F(n) has a uniform local structure around dead ends.

Load-bearing premise

The diagrams must faithfully encode every element of F(n) and the length read from the diagrams must coincide with the minimal word length in the standard generating set.

What would settle it

An explicit element of F(n) whose forest-diagram length differs from its known word length, or a dead-end element whose depth is not two.

Figures

Figures reproduced from arXiv: 2605.06739 by Mart\'in G\'omez Reynolds.

Figure 1
Figure 1. Figure 1: Infinite n-ary tree. Note that the homeomorphism ψn is congruent with the one defined by Burillo in [Bur, Section 1.4] for the case n = 2. Consider the real line as being subdivided by the integers, as shown below. -3 -2 -1 0 1 2 3 Define an n-adic subdivision of R as the result of subdividing finitely many of these intervals, of the form [k, k + 1] where k ∈ Z, into n-adic subintervals. We can represent a… view at source ↗
Figure 2
Figure 2. Figure 2: Forest diagrams of the n generators of F(n) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: An n-caret. Let us now examine how left-multiplying by the finite generating set of F(n) affects forest diagrams. The following proposition can be proven by direct computation. Note that we refer to the tree directly underneath the top pointer as the current tree. Proposition 2.8. Let f be a forest diagram for some f ∈ F(n). Then: (1) A forest diagram for x0f can be obtained by moving the top pointer of f … view at source ↗
Figure 4
Figure 4. Figure 4: Forest diagrams of xif for i ∈ {0, 1, 2, 3}. So left-multiplying by x0 moves the top pointer through the forest diagram, whereas x1, x2 and x3 build carets to the right of the top pointer. We say a forest diagram is reduced if it has no opposing pairs of n-carets. Reduction of n-ary forest diagrams works in the same way as the binary case. In particular, if f is a reduced forest diagram of f ∈ F(n) then th… view at source ↗
Figure 5
Figure 5. Figure 5: Correspondence between spaces and their enclosing n-carets. Example 3.6. Consider the element f ∈ F(3) with the forest diagram from the pre￾vious figure. Let us construct, in the most efficient way possible, the forest diagram of f by left-multiplying with the generating set {x0, x1, x2}. Let us note the number of times we must cross each marked space. We first construct the leftmost tree. To do this, we m… view at source ↗
Figure 6
Figure 6. Figure 6: Example of when to label marked spaces as N. We assign a weight to each marked space in the support of f according to the com￾bination of its top and bottom labels. See [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: An example of why order matters in Theorem 5.9. To finish this section, let us give a corollary from the previous theorem. Corollary 5.10. Let f ∈ F(n) with reduced forest diagram f. Then there exists a minimum-length word w for f with the following properties: (1) Each instance of xi for any i ∈ {1, . . . , n − 1} creates a caret in the top forest of f. (2) Each instance of x −1 i for any i ∈ {1, . . . , … view at source ↗
Figure 8
Figure 8. Figure 8: A typical dead end element f ∈ F(3). Note that, as the left space of f has label  L L  left-multiplying by x −1 0 will decrease the length, by Corollary 5.4. Also, left-multiplying f by x0 will decrease the length since the right space of f will now have top label L. Furthermore, since the right space of f has label  R R  , multiplying f by xi for any i ∈ {1, . . . , n−1} will decrease the length, by P… view at source ↗
read the original abstract

We extend the concept of two-way forest diagrams, introduced by Belk and Brown in 2003, to represent elements of $F(n)$ as a pair of infinite, bounded $n$-ary forests together with an order-preserving bijection of the leaves. This representation allows us to develop an alternative way to compute the length of an element of $F(n)$, distinct from the formula established by Fordham and Cleary in 2009. As an application of our length formula, we re-prove the existence of dead end elements in $F(n)$ and show that their depth is always two, first proved by Wladis in 2009.

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 manuscript extends Belk-Brown two-way forest diagrams to the generalized Thompson group F(n) by representing each element as a pair of infinite but bounded n-ary forests together with an order-preserving bijection of their leaves. From this representation the authors extract a length formula that they claim equals the word length with respect to the standard finite generating set; they then apply the formula to give an independent proof that dead-end elements exist in F(n) and that every such element has depth exactly two.

Significance. If the representation is bijective and the extracted length is provably minimal, the work supplies a new combinatorial calculus for F(n) that is independent of the Fordham-Cleary formula. The explicit diagram-to-word construction and the re-proof of the dead-end result are concrete strengths; the latter in particular demonstrates that the new length function can be used to recover known geometric facts without invoking the original Wladis argument.

major comments (2)
  1. [§4] §4, Theorem 4.3 and the surrounding discussion: the upper bound (any diagram yields a word of the stated length) is obtained by explicit multiplication rules on the forests, but the lower bound (no shorter word exists) is asserted via an invariant that is only sketched; an explicit verification that the invariant is preserved by every generator and that every diagram reduction is forced by the group operation is required before the length formula can be accepted as exact.
  2. [§5] §5, Proposition 5.1: the argument that dead-end depth is always two proceeds by exhibiting diagrams whose length cannot be reduced by right-multiplication by a generator; this relies on the correctness of the length formula established in §4, so the gap noted above is load-bearing for the dead-end claim as well.
minor comments (2)
  1. [Figure 2] Figure 2 and the caption on p. 7: the n-ary branching is drawn with varying edge thicknesses; a uniform convention or an additional legend would improve readability.
  2. [Definition 3.4] Definition 3.4: the phrase 'bounded n-ary forest' is introduced without an immediate formal definition; moving the definition one paragraph earlier would prevent the reader from having to infer the meaning from the subsequent examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive evaluation of its significance. We agree that the lower bound in the length formula requires a more explicit treatment and will revise the paper to supply the requested verifications. The responses to the major comments are given below.

read point-by-point responses

  1. Referee: [§4] §4, Theorem 4.3 and the surrounding discussion: the upper bound (any diagram yields a word of the stated length) is obtained by explicit multiplication rules on the forests, but the lower bound (no shorter word exists) is asserted via an invariant that is only sketched; an explicit verification that the invariant is preserved by every generator and that every diagram reduction is forced by the group operation is required before the length formula can be accepted as exact.

    Authors: We acknowledge that the lower bound argument in Theorem 4.3 relies on an invariant (the total caret count across both forests) whose invariance under generators and whose relation to diagram reductions were only sketched. In the revised manuscript we will insert a dedicated lemma that (i) enumerates the effect of right-multiplication by each standard generator on the pair of forests, (ii) shows that the invariant changes by at most the length of the generator, and (iii) proves that any diagram reduction step is forced by the group law (i.e., cannot be bypassed by a different sequence of generators). This will make the equality between the diagram length and the word length fully rigorous while leaving the upper-bound construction and the overall statement of the theorem unchanged. revision: yes

  2. Referee: [§5] §5, Proposition 5.1: the argument that dead-end depth is always two proceeds by exhibiting diagrams whose length cannot be reduced by right-multiplication by a generator; this relies on the correctness of the length formula established in §4, so the gap noted above is load-bearing for the dead-end claim as well.

    Authors: We agree that the dead-end result in Proposition 5.1 depends on the correctness of the length formula. Once the expanded verification of the lower bound is added to §4, the explicit diagrams constructed in §5 will satisfy the required minimality condition, confirming that right-multiplication by any generator cannot decrease the length. We will add a brief cross-reference in §5 to the new lemma in §4; no other modifications to the dead-end argument are needed. revision: yes

Circularity Check

0 steps flagged

No circularity: length formula derived from explicit diagram rules and faithful representation, independent of prior formulas

full rationale

The paper extends Belk-Brown forest diagrams to n-ary case via explicit construction of infinite bounded forests plus leaf bijection. It supplies diagram multiplication rules that construct words realizing the diagram length (upper bound) and argues via reduction invariants that this equals word length (lower bound). The resulting formula is presented as distinct from Fordham-Cleary and is used to re-prove dead-end elements independently of Wladis. No step reduces by definition, fitting, or self-citation chain to its own inputs; the representation is shown faithful by direct correspondence with group elements. This is the normal self-contained case for a diagrammatic proof in geometric group theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract was available; no explicit free parameters, invented entities, or non-standard axioms are mentioned.

axioms (2)
  • domain assumption Elements of F(n) are faithfully represented by the extended forest diagrams.
    The length formula and dead-end result rest on this representation being bijective and length-preserving.
  • domain assumption The standard generating set and word metric on F(n) are used.
    Length is computed with respect to the usual generators of F(n).

pith-pipeline@v0.9.0 · 5403 in / 1168 out tokens · 40734 ms · 2026-05-11T01:20:39.912113+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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    Burillo, José , title =

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