Pith. sign in

REVIEW 2 cited by

On the generation problem in Thompson's groups $F_n$

T0 review · reviewed 2026-06-28 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For every n≥2 the Higman-Thompson group Fn contains a maximal infinite-index subgroup isomorphic to F_{2n-1} that fixes no point of (0,1).

desk verdict The paper gives an explicit maximal infinite-index subgroup of F_n with no fixed points in (0,1), isomorphic to F_{2n-1}, plus sufficient conditions and an algorithm for the generation problem. read the letter →

arxiv 2606.00863 v1 pith:RI25V3CX submitted 2026-05-30 math.GR

Gili Golan , Eytan Sapir This is my paper · ORCID
classification math.GR
keywords Higman-ThompsongroupsgenerationproblemmaximalsubgroupsinfiniteindexThompson'sgroupFautomatacoreandclosurefixedpoints
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper develops sufficient conditions, based on the core and closure of subgroups of Fn together with associated automata, under which a subset generates the whole group Fn. It supplies an algorithm that checks these conditions when the subset is finite. The authors then apply the criteria to exhibit, inside each Fn for n at least 2, an explicit subgroup that is maximal, has infinite index, is isomorphic to F_{2n-1}, and fixes no point in the open interval (0,1). This construction answers a question of Aiello and Nagnibeda that arose from earlier work on maximal subgroups of Thompson's group F. A reader would care because the result gives a concrete existence statement and a practical method for settling generation questions in these groups.

What carries the argument

The core and closure of a subgroup of Fn, together with the finite automata that encode the generation relations among its elements.

What would settle it

An explicit larger proper subgroup of Fn properly containing the constructed H, or a direct verification that the constructed H fixes some point of (0,1).

Watch

Extended reading notes

Core claim

The central claim is that the core-and-closure conditions are sufficient to prove both that a certain explicitly described subgroup H of Fn is proper and that H is maximal; the same conditions also establish that H is isomorphic to F_{2n-1} and that its action on (0,1) has empty fixed-point set. The construction works uniformly for every n≥2 and is verified by checking the automata associated to the core and closure of H.

Load-bearing premise

The core-and-closure conditions developed in the paper are sufficient to certify both the generation property and the maximality of the constructed subgroup.

Editorial extensions

If this is right

  • The generation problem for finite subsets of Fn is decidable whenever the core-and-closure conditions apply.
  • Each Fn for n≥2 admits at least one maximal subgroup of infinite index whose fixed-point set in (0,1) is empty.
  • The constructed maximal subgroup is isomorphic to the Higman-Thompson group F_{2n-1}.
  • The same core-and-closure technique produces further examples of maximal subgroups in the same groups.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The existence of these pointwise-fixed-point-free maximal subgroups suggests that the lattice of subgroups of Fn is more varied than the examples previously obtained by fixing points.
  • The automata-based verification method could be adapted to decide generation questions inside other Thompson-like groups.
  • Whether every maximal infinite-index subgroup of Fn is isomorphic to some F_m remains open and could be tested by applying the same conditions to other candidate subgroups.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 0 minor

Summary. The paper develops sufficient conditions, based on the core and closure of subgroups together with associated automata, for a finite or infinite subset X of the Higman-Thompson group F_n to generate the whole group. It supplies an algorithm that certifies these conditions when X is finite. As an application it constructs, for each n ≥ 2, an explicit subgroup isomorphic to F_{2n-1} that is maximal of infinite index in F_n and fixes no point of (0,1), thereby answering a question of Aiello and Nagnibeda.

Significance. The work supplies new, algorithmically verifiable criteria for generation in the Higman-Thompson groups and gives an explicit, parameter-free construction that resolves an open question on maximal subgroups. The algorithmic verification of the core-and-closure conditions and the concrete isomorphism type of the constructed subgroup are concrete strengths that can be checked independently.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and supportive report, including the clear recommendation to accept the manuscript. The absence of any major comments means there are no specific points requiring a point-by-point response or revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is self-contained

full rationale

The paper develops sufficient conditions on the core and closure of subgroups (with associated automata) to certify generation of Fn, states an algorithm to check these for finite sets, and applies them to an explicit construction of a subgroup isomorphic to F_{2n-1}. These conditions are derived from the established structure of Thompson groups and automata theory; the maximality and fixed-point-free properties follow directly from the conditions holding for the constructed set. No step reduces by definition or construction to a fitted parameter, self-citation chain, or renamed input; the central claims rest on independent verification rather than circular reduction.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; full manuscript would be required to audit the background assumptions on automata and subgroup cores.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On the generation problem in Thompson's groups $F_n$." pith.science (2026). https://pith.science/paper/RI25V3CX

@misc{pith2026260600863,
  author       = {Pith},
  title        = {Pith review of: On the generation problem in Thompson's groups $F_n$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RI25V3CX}},
  note         = {Machine review of arXiv:2606.00863}
}
abstract

We study the generation problem in the Higman-Thompson groups $F_n$ via the core and closure of subgroups of $F_n$ and associated automata. We give sufficient conditions for a subset $X \subseteq F_n$ to generate $F_n$, and provide an algorithm which verifies these conditions when $X$ is finite. As an application, we answer a question of Aiello and Nagnibeda, motivated by Savchuk's problem on maximal subgroups of Thompson's group $F$. Specifically, we show that for every $n\geq 2$, the Higman-Thompson group $F_n$ contains a maximal subgroup of infinite index which fixes no point of $(0,1)$. The subgroup we construct is isomorphic to $F_{2n-1}$.

Figures

Figures reproduced from arXiv: 2606.00863 by the authors.

Figure 2.1
Figure 2.1. A ternary tree. Edges are labeled by ’0’,’1’,’2’. Each vertex corresponds to a triadic [PITH_FULL_IMAGE:figures/full_fig_p006_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. The top and bottom tree-diagrams correspond to the same element of [PITH_FULL_IMAGE:figures/full_fig_p007_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. The core of F3, L(F3). Definition 2.16. Let H be a subgroup of Fn. We define the closure of H, denoted by Cl(H), as the subgroup of Fn of all elements accepted by L(H). The closure operation satisfies standard properties of a closure operator. Namely, if H is a subgroup of Fn, then H ≤ Cl(H) and Cl(Cl(H)) = Cl(H); furthermore, if H1 ≤ H2 are subgroups of Fn, then Cl(H1) ≤ Cl(H2). Let H be a subgroup of Fn. A functio… view at source ↗
Figures from the paper (11 more)
Figure 4.4
Figure 4.4. Figure 4.4: An example of a rooted tree-automaton Ar, the path tree TAr and a minimal rooted tree T min Ar associated with Ar. Each vertex in those trees, corresponding to a path p in the automaton, is labeled by the end vertex of p. Every caret in each of the trees, rooted in s…
Figure 5.5
Figure 5.5. Figure 5.5: An image describing a minimal tree associated with the core of [PITH_FULL_IMAGE:figures/full_fig_p033_5_5.png]
Figure 5.6
Figure 5.6. Figure 5.6: An illustration of minimal tree associated with the core of [PITH_FULL_IMAGE:figures/full_fig_p034_5_6.png]
Figure 5.7
Figure 5.7. Figure 5.7: The relation r = f ·a1 · · · a2n−3 ·g was replaced by the relations r = f0 ·an · · · a2n−3 ·g, and f0 = f · a1 · · · an−1. Next, we construct a basic splitting of the previous presentation by adding a new generator denoted by cn−1, removing the relation f = f · a1 · …
Figure 5.8
Figure 5.8. Figure 5.8: The relation f = f · a1 · · · a2n−3 · a0 was replaced by the relations f = f · a1 · · · an−2 · cn−1, and cn−1 = an−1 · · · a2n−3 · a0 Next, we construct a basic splitting of the previous presentation by adding a new generator denoted by c0, removing the relation g = …
Figure 5.9
Figure 5.9. Figure 5.9: The relation g = a0 · · · a2n−3 · g was replaced by the relations g = c0 · an · · · a2n−3 · g, and c0 = a0 · · · an−1. We now construct a splitting of the previous presentation by adding new generators denoted by b0, . . . , b2n−3. For each i ∈ {0, . . . , 2n − 3}, w…
Figure 5.10
Figure 5.10. Figure 5.10: The relations ai = ai · ai+1 · · · ai , where i ∈ {0, . . . , 2n − 3} were removed, and new relations and generators were added. We conclude the sequence of splittings here and proceed to apply foldings of type 2 to the 37 [PITH_FULL_IMAGE:figures/full_fig_p037_5_10.png]
Figure 5.11
Figure 5.11. Figure 5.11: c0, b0, bn−1, . . . , b2n−3 are identified by foldings of type 2 to a vertex denoted by c0. cn−1, b1, . . . , bn−2 are identified by foldings of type 2 to a vertex denoted by cn−1. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_5_11.png]
Figure 5.12
Figure 5.12. Figure 5.12: This figure illustrates a minimal tree of the [PITH_FULL_IMAGE:figures/full_fig_p041_5_12.png]
Figure 5.13
Figure 5.13. Figure 5.13: Tree-diagrams of the elements h0, . . . , hn−1. Each vertex in those tree-diagrams is labeled by its appropriate label in TAr . Note that every vertex is labeled, and that the end￾vertices of each pair of branches share a label, so the elements are accepted by Ar. W…
Figure 5.14
Figure 5.14. Figure 5.14: The graph G(Ar). The nodes represent possible identifications, and the edges represent following identifications, after foldings of type 1. Since the identifications of vertices that follow from the identification ai = ai+n−1 do not depend on i for all i ∈ {1, . . .…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Higman--Thompson groups $F_n$ all the way down

    math.GR 2026-07 accept novelty 8.0 of 10

    Every Higman–Thompson group Fn admits a chain of maximal infinite-index copies of itself with trivial intersection, realized by semi-synchronizing transducers.

  2. Irreducible fast sets of bump homeomorphisms generate copies of Thompson's groups $F_n$

    math.GR 2026-07 accept novelty 7.5 of 10

    Irreducible geometrically fast sets of n positive bumps generate groups isomorphic to the n-ary Thompson group F_n for every n≥2.

Reference graph

Works this paper leans on

22 extracted references · 6 canonical work pages · cited by 2 Pith papers

  1. [1]

    Aiello and T

    V. Aiello and T. Nagnibeda,On the oriented Thompson subgroup ⃗F3 and its relatives in higher Brown-Thompson groups, Journal of Algebra and Its Applications,21(07), 2250139 (2022)

    2022

  2. [2]

    Aiello and T

    V. Aiello and T. Nagnibeda, On the 3-colorable subgroupFand maximal subgroups of Thompson’s groupF,Annales de l’Institut Fourier73(2023), no. 2, 783–828. doi:10.5802/aif.3555

    work page doi:10.5802/aif.3555 2023

  3. [3]

    Bleak, M

    C. Bleak, M. G. Brin, M. Kassabov, J. Tatch Moore, and M. C. B. Zaremsky,Groups of fast homeomorphisms of the interval and the ping-pong argument, J. Comb. Algebra3(2019), no. 1, 1–40.doi:10.4171/JCA/25

    work page doi:10.4171/jca/25 2019

  4. [4]

    Belk and B

    J. Belk and B. Forrest,Divergence in the Brown–Thompson groups, Transform. Groups (2023), published online.doi:10.1007/s00031-023-09839-8

    work page doi:10.1007/s00031-023-09839-8 2023

  5. [5]

    Belk and L

    J. Belk and L. Stott,Pseudo-F4 is isomorphic toF4, arXiv:2303.16868

    work page arXiv

  6. [6]

    78 (2008), no

    CollinBleak, Ageometricclassificationofsomesolvablegroupsofhomeomorphisms,Journal of the London Mathematical Society, Series 2, vol. 78 (2008), no. 2, pp. 352–372

    2008

  7. [7]

    K. S. Brown and R. Geoghegan,An infinite-dimensional torsion-freeFP∞ group, Invent. Math.,77(1984), 367–381

    1984

  8. [8]

    K. S. Brown,Finiteness properties of groups, Journal of Pure and Applied Algebra,44 (1987)

    1987

Show all 22 references
  1. [9]

    Burillo, K.-U

    J. Burillo, K.-U. Bux and B. Nucinkis,Cohomological and metric properties of groups of homeomorphisms ofR, Oberwolfach Reports15(2018), 1579–1633

    2018

  2. [10]

    J. W. Cannon, W. J. Floyd, and W. R. Parry. Introductory notes on Richard Thompson’s groups.L’Enseignement Mathématique, 42(3):215–256, 1996

    1996

  3. [11]

    Golan-Polak,The generation problem in Thompson groupF, Memoirs of the AMS,292, no

    G. Golan-Polak,The generation problem in Thompson groupF, Memoirs of the AMS,292, no. 1451 (2023)

    2023

  4. [12]

    Golan-Polak,On maximal subgroups of Thompson’s groupF, Groups, Geometry, and Dynamics 19 (3), 797-860.doi:10.4171/GGD/795

    G. Golan-Polak,On maximal subgroups of Thompson’s groupF, Groups, Geometry, and Dynamics 19 (3), 797-860.doi:10.4171/GGD/795

    doi

  5. [13]

    Golan and M

    G. Golan and M. Sapir,On subgroups of R. Thompson groupF, Trans. Amer. Math. Soc. 369(2017), 8857–8878

    2017

  6. [14]

    Golan-Polak and M

    G. Golan-Polak and M. Sapir,On closed subgroups of the R. Thompson groupF, Israel J. Math.267(2025), 35–84.doi:10.1007/s11856-024-2692-z 50

    2025 doi

  7. [15]

    V. S. Guba and M. V. Sapir,Diagram groups, Memoirs Amer. Math. Soc.,130(620), 1997

    1997

  8. [16]

    V. S. Guba and M. V. Sapir,On subgroups of R. Thompson’s groupFand other diagram groups, Sb. Math. (English),190(8) (1999), 1077–1130

    1999

  9. [17]

    8, Australian National University, Canberra, 1974

    Graham Higman,Finitely Presented Infinite Simple Groups, Notes on Pure Mathematics, no. 8, Australian National University, Canberra, 1974

    1974

  10. [18]

    R. C. Lyndon and P. E. Schupp,Combinatorial Group Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 89, Springer–Verlag, Berlin, 1977

    1977

  11. [19]

    Savchuk,Some graphs related to Thompson’s groupF, In: Combinatorial and Geometric Group Theory, 279–296, Trends Math., Birkhäuser/Springer, Basel, 2010

    D. Savchuk,Some graphs related to Thompson’s groupF, In: Combinatorial and Geometric Group Theory, 279–296, Trends Math., Birkhäuser/Springer, Basel, 2010

    2010

  12. [20]

    Savchuk,Schreier graphs of actions of Thompson’s groupFon the unit interval and on the Cantor set, Geom

    D. Savchuk,Schreier graphs of actions of Thompson’s groupFon the unit interval and on the Cantor set, Geom. Dedicata,175(2015), 355–372

    2015

  13. [21]

    Stein,Groups of piecewise linear homeomorphisms, Trans

    M. Stein,Groups of piecewise linear homeomorphisms, Trans. Amer. Math. Soc.332(1992), no. 2, 477–514

    1992

  14. [22]

    Wladis,Thompson’s groupF(n)is not minimally almost convex, New York J

    C. Wladis,Thompson’s groupF(n)is not minimally almost convex, New York J. Math., 13(2007), 319–326.https://nyjm.albany.edu/j/2007/13-19.pdf Gili Golan Department of Mathematics, Ben-Gurion University of the Negev, golangi@bgu.ac.il Eytan Sapir Department of Mathematics, Ben-Gu...

    2007

Pith tools

Reviewed June 28, 2026 · model on record in the stance chip above.