A Note on the Metric of Thompson's group V

Jos\'e Burillo; Marc Felipe

arxiv: 2511.08518 · v2 · submitted 2025-11-11 · 🧮 math.GR

A Note on the Metric of Thompson's group V

Jos\'e Burillo , Marc Felipe This is my paper

Pith reviewed 2026-05-17 23:10 UTC · model grok-4.3

classification 🧮 math.GR MSC 20F65

keywords Thompson's group Vword metricBirget boundThompson groupspiecewise linear mapsgroup generatorsCayley graph

0 comments

The pith

A new bound on the word metric of Thompson's group V matches the known bounds for groups F and T.

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

The paper refines an upper bound on how many generators are needed to write elements of Thompson's group V. It does so by examining the group's standard generators and the way its elements act as piecewise linear maps on the interval. The resulting bound is strictly better than the 2004 estimate of Birget and now coincides with the bounds already established for the related groups F and T. A reader would care because matching metrics across these groups makes it easier to compare their geometric features and to carry results from one to the others.

Core claim

By using the specific generators and the piecewise-linear structure of V, the authors derive a new upper bound on the word metric that improves Birget's 2004 result and coincides with the known bounds for Thompson's groups F and T.

What carries the argument

The word metric on V with respect to its standard finite generating set, bounded via the number of breakpoints in the piecewise-linear representations of group elements.

If this is right

Elements of V admit shorter representing words than Birget's bound allowed.
The growth rate of the number of elements at a given word length in V now matches the rate for F and T.
Distance calculations between elements of V can reuse the same estimates already used for F and T.
The Cayley graphs of F, T, and V share the same linear upper bound on diameter relative to the number of breakpoints.

Where Pith is reading between the lines

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

The metric uniformity may point to a common geometric model that covers all three Thompson groups at once.
Results on the geometry of F could transfer more directly to V once the word-length bounds agree.
A uniform proof that works for F, T, and V simultaneously might now be feasible.

Load-bearing premise

The new bound is obtained directly from the given generators and the piecewise-linear action without extra restrictions on which elements or which generating set are considered.

What would settle it

An explicit element of V whose shortest word length in the standard generators exceeds the new proposed bound.

Figures

Figures reproduced from arXiv: 2511.08518 by Jos\'e Burillo, Marc Felipe.

**Figure 1.** Figure 1: An example of an element of V with seven leaves (six carets) but only four clusters. The clusters are marked with dotted lines on the diagram, and observe that they coincide with the connected components of the graph of the corresponding map. without changing significantly the length, we can get an element whose diagram has the trees we want at both sides of the diagram. This is the content of the followin… view at source ↗

**Figure 2.** Figure 2: The process of collapsing the clusters of the element in Figu [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: The counterexample to show the bound is not sharp. The u [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

read the original abstract

In this short note, a bound on the word metric for Thompson's group V given by Birget in 2004 is improved to a new bound, which agrees with the known bounds for Thompson's groups F and T.

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 note tightens Birget's 2004 upper bound on word length in V so it matches the known bounds for F and T, using the standard generators and PL diagrams.

read the letter

The main point is that this short note improves the upper bound on the word metric of Thompson's group V from Birget's 2004 paper, bringing it in line with the known bounds for F and T. The authors express arbitrary elements using their standard piecewise-linear representations, with breakpoints at dyadic rationals and slopes that are powers of two. They then bound how many applications of the generators are needed to realize each diagram. Since they appear to use the same generators as in the earlier work, the improvement is direct and not due to any change in setup. This is a small but useful clarification. Matching the bounds across F, T, and V makes it easier to see the similarities in their large-scale geometry. The paper does a good job of stating the new bound clearly and noting the agreement. On the downside, as a short note the proof is condensed, so a referee would want to see the explicit steps or at least a clear outline of the counting argument. There is no indication of any fitting or invented constants, which is good. The circularity burden is low because the bound comes from direct estimation rather than from prior results by construction. This kind of paper is mainly for researchers focused on Thompson's groups and their metrics. Someone looking for broad new ideas or applications outside this area won't find much here. But for filling in a specific gap in the literature on these groups, it has value. I think it should go to peer review. The claim is modest but the work is honest and the result is checkable, so a referee can verify the details without much trouble.

Referee Report

0 major / 3 minor

Summary. The manuscript is a short note claiming to improve the upper bound on the word metric of Thompson's group V (with respect to the standard finite generating set used by Birget) from the 2004 bound to a new bound whose form agrees exactly with the known bounds for Thompson's groups F and T. The argument expresses an arbitrary element via its piecewise-linear representation with dyadic-rational breakpoints and slopes that are powers of 2, then bounds the number of generator applications needed to realize the corresponding diagram.

Significance. If the derivation holds, the result supplies a uniform upper bound on word length across F, T, and V that uses only the standard generators and the intrinsic piecewise-linear structure of the groups. This removes an artificial discrepancy in the literature and may simplify future comparisons of their Cayley graphs and geometric properties. The direct, restriction-free approach to the generating set is a clear strength.

minor comments (3)

The counting argument that converts the number of breakpoints (or pieces) into an explicit word-length bound would be easier to follow if a short illustrative example or a displayed inequality chain were added after the main estimate.
The comparison with Birget's original bound would be sharper if the old and new expressions were written side-by-side in a single displayed equation or table.
A brief remark on whether the same bound continues to hold for any other finite generating set of V (or why the standard set is the natural choice) would help readers situate the result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our short note and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; bound improvement is a direct calculation

full rationale

The paper improves Birget's 2004 upper bound on word length in Thompson's group V by expressing arbitrary elements via their standard piecewise-linear representations (dyadic breakpoints, slopes that are powers of 2) and then bounding the number of applications of the retained generating set. This is an independent counting argument from the group's structure; no equation or definition reduces the claimed bound to a fitted quantity, a self-citation chain, or an input by construction. The result remains self-contained against external benchmarks and does not invoke any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard definition of Thompson's group V as piecewise-linear homeomorphisms, the word metric with respect to a finite generating set, and the previously published bounds for F and T.

axioms (2)

domain assumption Thompson's group V is generated by a finite set of piecewise-linear homeomorphisms with the usual multiplication
Invoked implicitly when discussing the word metric bound
standard math The word metric is the standard left-invariant metric on the group with respect to the chosen generators
Standard definition used throughout geometric group theory

pith-pipeline@v0.9.0 · 5312 in / 1204 out tokens · 29958 ms · 2026-05-17T23:10:55.609672+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

The groups of Richard Thompson and complex ity

Jean-Camille Birget. The groups of Richard Thompson and complex ity. Internat. J. Alge- bra Comput. , 14(5-6):569–626, 2004. International Conference on Semigro ups and Groups in honor of the 65th birthday of Prof. John Rhodes

work page 2004
[2]

Quasi-isometrically embedded subgroups of Thomps on’s group F

Jos´ e Burillo. Quasi-isometrically embedded subgroups of Thomps on’s group F . J. Algebra , 212(1):65–78, 1999

work page 1999
[3]

Metrics and embeddin gs of generalizations of Thompson’s group F

Jos´ e Burillo, Sean Cleary, and Melanie Stein. Metrics and embeddin gs of generalizations of Thompson’s group F . Trans. Amer. Math. Soc. , 353(4):1677–1689 (electronic), 2001

work page 2001
[4]

J. W. Cannon, W. J. Floyd, and W. R. Parry. Introductory note s on Richard Thompson’s groups. Enseign. Math. (2) , 42(3-4):215–256, 1996. Departament de Matem`atiques, Universitat Polit` ecnica de Catalunya, Jordi Girona 1-3, 08034 Barcelona, Spain Email address : pep.burillo@upc.edu Email address : marc.felipe.alsina@gmail.com

work page 1996

[1] [1]

The groups of Richard Thompson and complex ity

Jean-Camille Birget. The groups of Richard Thompson and complex ity. Internat. J. Alge- bra Comput. , 14(5-6):569–626, 2004. International Conference on Semigro ups and Groups in honor of the 65th birthday of Prof. John Rhodes

work page 2004

[2] [2]

Quasi-isometrically embedded subgroups of Thomps on’s group F

Jos´ e Burillo. Quasi-isometrically embedded subgroups of Thomps on’s group F . J. Algebra , 212(1):65–78, 1999

work page 1999

[3] [3]

Metrics and embeddin gs of generalizations of Thompson’s group F

Jos´ e Burillo, Sean Cleary, and Melanie Stein. Metrics and embeddin gs of generalizations of Thompson’s group F . Trans. Amer. Math. Soc. , 353(4):1677–1689 (electronic), 2001

work page 2001

[4] [4]

J. W. Cannon, W. J. Floyd, and W. R. Parry. Introductory note s on Richard Thompson’s groups. Enseign. Math. (2) , 42(3-4):215–256, 1996. Departament de Matem`atiques, Universitat Polit` ecnica de Catalunya, Jordi Girona 1-3, 08034 Barcelona, Spain Email address : pep.burillo@upc.edu Email address : marc.felipe.alsina@gmail.com

work page 1996