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

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Pith's one-line read Every Higman–Thompson group Fn contains a descending chain of copies of itself, each maximal of infinite index in the one above, with trivial intersection.

desk verdict Explicit self-embedding maximal chains for every Fn, with a reusable transducer–core calculus that also settles ⃗F3≅F4. read the letter →

arxiv 2607.04038 v1 pith:CCE5P4UK submitted 2026-07-04 math.GR

Gili Golan This is my paper · ORCID
classification math.GR MSC 20F6520E2837B1020F10
keywords Higman–ThompsongroupsThompsongroupFmaximalsubgroupssemi-synchronizingtransducersStallingscoresJonesorientedsubgroupCantor-spaceconjugacydiagram
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 shows that each Higman–Thompson group Fn (n≥2) is “Fn all the way down”: it admits a nested sequence Fn = H0 > H1 > H2 > ··· in which every Hi is isomorphic to Fn, the intersection of the whole chain is the identity, and the only subgroups of Fn that contain Hi are Hi itself and the finitely many larger groups in the chain. In particular each step Hi+1 is a maximal subgroup of infinite index inside Hi. The construction rests on a dictionary between finite transducers (machines that rewrite infinite sequences letter by letter) and the Stallings-type cores that encode closed subgroups of Fn. That dictionary turns conjugation by a carefully chosen rational homeomorphism into an algorithmic computation of cores, so the overgroups of each Hi can be read off by inspecting finite automata. Along the way the same tools identify Jones’s ternary oriented subgroup of F3 with F4 and show that every previously known infinite-index maximal subgroup of Thompson’s F that acts minimally on the unit interval is isomorphic to some Higman–Thompson group.

What carries the argument

A bridge between finite semi-synchronizing transducers and Stallings 2-cores of closed subgroups of Fn that turns conjugation of a closed subgroup by a rational Cantor-space homeomorphism into an explicit finite-automaton computation of the conjugate core.

What would settle it

Exhibit, for some n, either a proper overgroup of one of the constructed Hi that is not equal to any Hj, or a maximal infinite-index subgroup of F that acts minimally on (0,1) and is not isomorphic to any Higman–Thompson group.

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Extended reading notes

Core claim

For every n≥2 there exists a chain Fn=H0>H1>H2>··· of subgroups, each isomorphic to Fn, with trivial total intersection, such that every subgroup of Fn containing Hi is one of Hi,Hi−1,…,H0; consequently each Hi+1 is maximal of infinite index in Hi.

Load-bearing premise

The step from closed overgroups to all overgroups leans on generation criteria for F and for Fn that guarantee a closed subgroup with full abelianization and the right number of inner core states must be the whole group.

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Editorial analysis

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

Referee Report

0 major / 4 minor

Summary. The paper proves that for every n≥2 the Higman–Thompson group Fn admits a descending chain Fn=H0>H1>H2>⋯ of subgroups, each isomorphic to Fn, with trivial intersection, such that every subgroup of Fn containing Hi is one of Hi,...,H0; in particular each Hi+1 is maximal of infinite index in Hi (Theorem 1.1 / Theorem 7.30). The argument rests on a characterization of Cantor-space conjugators of Fn into Fm as order-preserving or order-reversing rational homeomorphisms whose minimal transducers are semi-synchronizing (Theorem 4.11), together with a pullback/forward (and geometric) construction that computes cores of conjugated closed subgroups from transducers (Section 5). Applications include →F3≅F4 and the realization of all known minimally acting infinite-index maximal subgroups of F as Higman–Thompson groups. The binary chain is obtained from powers of an explicit four-state transducer; the general-n chain is obtained by residue inflation of that transducer followed by a compatible twisting argument.

Significance. The main theorem settles the natural boundary case of the program of constructing infinite-index maximal subgroups of Fn that are not point stabilizers: one obtains maximal copies of Fn inside itself, and even infinite descending chains of such maximals with trivial intersection. The semi-synchronizing conjugator criterion and the transducer–core bridge are of independent interest; they give an algorithmic way to conjugate finitely generated closed subgroups and immediately yield →F3≅F4 (answering Aiello) and the identification of all currently known minimally acting infinite-index maximals of F with Higman–Thompson groups. The constructions are fully explicit (transducers, cores Ai, residue inflation, twisting), so the results are checkable rather than purely existential. The work also frames a clean open problem (Problem 1.4) and situates related forthcoming results on fast groups and diagram groups.

minor comments (4)
  1. [Sections 5–7] The manuscript is long and dense; a short roadmap at the start of Sections 5–7 (what is proved, what is only used later) would help the reader navigate the geometric determinization and the residue-inflation/twisting steps.
  2. [Section 2 / Section 5] Notation for cores, folded quotients, and local actions is heavy; a one-page notation index (or a brief reminder table at the start of Section 5) would reduce the need to flip back to Section 2.
  3. [Figures 1, 2, 6–9, 10–15] Figures 1, 2, 6–9 and the geometric subdivision figures are essential; ensuring that edge labels (input|output) and state renamings (e.g., Pt,Dt o a2t,a2t+1) remain legible in the final layout would improve readability.
  4. [Introduction / §2.10] The dependence on the generation criteria (Theorems 2.22 and 2.23 from prior work) is correctly used after the abelianization and closure hypotheses are checked for the specific conjugated subgroups; a one-sentence pointer in the introduction that those hypotheses are verified in Lemmas 6.5 and 2.26 (and the parallel n-ary statements) would make the logical structure even clearer.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: maximality is deduced from independently computed cores plus prior generation lemmas whose hypotheses are checked for the constructed conjugates.

  1. self citation load bearing [§2.10 Theorems 2.22–2.23; Lemma 2.26; Theorems 6.6 and 7.26]
    "We shall use the following generation criterion for Thompson’s group F from [20]. Theorem 2.22 (Generation theorem for F). Let H≤F. Then H=F if and only if H[F,F]=F and [F,F]≤Cl(H). … We shall also use the sufficient generation criterion for Fn from [22, Theorem 3.19]."

    Removing closedness to obtain that every (not merely closed) overgroup of Hi is some Hj relies on generation theorems proved in the author’s earlier papers. This is load-bearing for the full strength of Theorem 1.1, but the cited statements are general criteria whose hypotheses are independently verified for the conjugated subgroups constructed here; they do not encode the chain itself. Mild self-citation, not definitional circularity.

full rationale

The chain Hi = F^{φ^i} (and the n-ary analogues Si) is defined by an explicit semi-synchronizing transducer and its powers; the cores Ai are computed inductively by the forward/geometric construction of §5–6, and closed overgroups of Hi are classified by quotients of Ai (Lemma 6.3, Theorem 6.4) without presupposing maximality. The only load-bearing external inputs are the generation criteria for F and Fn (Theorems 2.22–2.23 from the author’s prior work) and the automatic closedness of infinite-index maximals in F ([20]). Those results are general lemmas with stated hypotheses; the paper verifies the hypotheses (full cores with the right number of inner states, full abelianization image, Lemma 6.5, Lemma 2.26) rather than assuming the target chain. Semi-synchronization, residue inflation, and the pullback/forward automata are developed from first principles in this paper. No quantity is fitted and re-predicted, no uniqueness theorem is imported to forbid alternatives by fiat, and no known empirical pattern is merely renamed. Score 1 reflects ordinary self-citation of prior technical lemmas that are not themselves restatements of Theorem 1.1.

Assumptions & free parameters 0 free parameters · 6 assumptions · 4 invented entities

Pure group-theory paper. Load-bearing background is standard Thompson/Higman–Thompson structure, Stallings-type cores from the author’s prior work, generation criteria for F and Fn, and the Bleak–Olukoya transducer framework for automorphisms. The paper’s own inventions are the semi-synchronizing condition, the pullback/forward core constructions, residue inflation, and compatible twisting. No numerical free parameters.

assumptions (6)
  • standard math Standard structure of Higman–Thompson groups Fn (tree diagrams, abelianization Zn, simple derived subgroup, orbit criterion via σn).
    Used throughout §§2–3; classical Brown/Cannon–Floyd–Parry material.
  • domain assumption Closed subgroups of Fn are exactly diagram groups of folded tree-automata; cores have the existence property (Guba–Sapir / Golan prior work).
    §2.6–2.8; foundation of the overgroup-via-quotient analysis.
  • domain assumption Generation theorem for F: H=F iff H[F,F]=F and [F,F]≤Cl(H) (Theorem 2.22).
    Used in Lemma 2.26 and Theorem 6.6 to remove closedness.
  • domain assumption Sufficient generation criterion for Fn via semi-cores and fixed-point slope conditions (Theorem 2.23 from [22]).
    Invoked for the n-ary chain overgroup control in §7.
  • standard math Rubin-type reconstruction for locally moving groups (Brum–Matte Bon–Rivas–Triestino) and McCleary–Rubin normalizer identification.
    §3 and Corollary 4.12; converts abstract maximal copies into interval conjugacies.
  • standard math Rational homeomorphisms of Cantor space are exactly those with finitely many local actions; minimization and inverse transducers exist (Grigorchuk–Nekrashevych–Sushchanskii).
    §2.13–2.15; background for the conjugator characterization.
invented entities (4)
  • Semi-synchronizing transducers independent evidence
    purpose: Characterize homeomorphisms conjugating Fn into Fm; replace full bi-synchronization used for Gn,r / Tn,r.
    Definition 4.10; boundary-ray stabilization plus inner synchronization by σn class. Independent mathematical definition with explicit examples.
  • Pullback and forward core automata (and geometric determinization) independent evidence
    purpose: Algorithmically compute cores of conjugated closed subgroups H^ψ ∩ Fm.
    §5; bridges transducers to Stallings 2-cores. Verifiable on finite examples in the paper.
  • Residue inflation of binary transducers to n-ary transducers independent evidence
    purpose: Lift the binary chain construction to general Fn.
    §7.1; systematic embedding of binary cells into residue-zero states of C(Fn).
  • Compatible twisting of the standard chain Si by coherent inner conjugations
    purpose: Force trivial intersection while preserving the overgroup lattice.
    §7 final step / Proposition 7.29; ad hoc to this paper but fully defined.

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Pith. "Pith review of Higman--Thompson groups $F_n$ all the way down." pith.science (2026). https://pith.science/paper/CCE5P4UK

@misc{pith2026260704038,
  author       = {Pith},
  title        = {Pith review of: Higman--Thompson groups $F_n$ all the way down},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CCE5P4UK}},
  note         = {Machine review of arXiv:2607.04038}
}
abstract

We prove that for every $n\ge 2$ the Higman--Thompson group $F_n$ has a maximal subgroup of infinite index isomorphic to itself. In fact, we construct a chain of subgroups $F_n=H_0>H_1>H_2>\cdots$, all isomorphic to $F_n$ and with trivial intersection, such that for every $i$ the only subgroups of $F_n$ containing $H_i$ are $H_i,H_{i-1},\ldots,H_0=F_n$; in particular, each $H_{i+1}$ is maximal in $H_i$. We prove that for all $n\ge m\ge 2$, every closed maximal subgroup of $F_m$ isomorphic to $F_n$ arises from a homeomorphism between the $n$-ary and $m$-ary Cantor spaces given by a finite semi-synchronizing transducer--a variation of the synchronizing transducers of Bleak, Cameron, Maissel, Navas and Olukoya. We characterize the homeomorphisms of Cantor spaces conjugating $F_n$ into $F_m$ as the order-preserving or order-reversing rational homeomorphisms whose minimal transducer is semi-synchronizing. At the heart of the paper is a machinery bridging transducers and Stallings $2$-cores of subgroups, which reduces the conjugation of finitely generated closed subgroups by such homeomorphisms to an algorithmic procedure. As applications, we prove that Jones' ternary oriented subgroup $\vec F_3\le F_3$ is isomorphic to $F_4$, answering questions of Aiello, and that all known maximal subgroups of infinite index of Thompson's group $F$ which act minimally on $(0,1)$ are isomorphic to Higman--Thompson groups. That raises the problem of whether all maximal subgroups of infinite index of $F$ which act minimally on $(0,1)$ are isomorphic to Higman--Thompson groups. We briefly discuss related results regarding fast groups of homeomorphisms and maximal subgroups of Thompson groups.

Figures

Figures reproduced from arXiv: 2607.04038 by the authors.

Figure 1
Figure 1. The minimal binary semi-synchronizing transducer defining [PITH_FULL_IMAGE:figures/full_fig_p036_1.png] view at source ↗
Figure 2
Figure 2. The minimal (4, 3)-transducer TJ representing ψJ . Edge labels are input–output labels; the full transition-output table is (J1). Let J be the following full ternary tree-automaton. Its states are ρ, ℓ, r0, r1, ζ00, ζ01, ζ10, ζ11, 52 [PITH_FULL_IMAGE:figures/full_fig_p052_2.png] view at source ↗
Figure 3
Figure 3. The core B = C(B) of the Brin–Navas subgroup B. Define the binary minimal initial transducer T ψ with state set {ρ, ℓ, r, a, b, c, d}, initial state ρ, and transition-output table 0 1 ρ 0 : ℓ 1 : r ℓ 0 : ℓ 1 : a r 0 : b 1 : r a 0 : a 1 : c b 0 : c 1 : b c 00 : a ε : d d 01 : c 1 : b. Here the entry w : t in the column labelled i means that, on input i, the transducer outputs w and moves to the state t. The transduce… view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: The transducer T ψ. Edge labels are input-output labels. be an order-preserving or order-reversing homeomorphism, and let T ψ = (Sψ, tψ, oψ, sψ) be its minimal (n, m)-transducer. Let Kn = C(Fn) be the standard core from Lemma 2.21. Assume that, after forgetting outputs…
Figure 5
Figure 5. Figure 5: The only nontrivial geometric step. Contracting the unique [PITH_FULL_IMAGE:figures/full_fig_p071_5.png]
Figure 6
Figure 6. Figure 6: Applying the forward construction to (TK1 , C(F3)), and then folding, gives the core L1. Hence K1 = F ψK1 3 ∼= F3. The transitive subgroup K3. The subgroup from [19, Proposition 10.15] is K3 = ⟨x0, x1x2x −3 1 , x1x2x3x −3 2 x −1 1 ⟩. It is a maximal subgroup of F of in…
Figure 7
Figure 7. Figure 7: The core of K3 from [19, Proposition 10.15] and a binary transducer realizing it by the geometric construction with 12. The Aiello–Nagnibeda examples Aiello and Nagnibeda work inside the rectangular subgroup K(2,2) = {f ∈ F | log2 f ′ (0), log2 f ′ (1) ∈ 2Z} ∼= F. They…
Figure 8
Figure 8. Figure 8: The core of M0 and a minimal (3, 2)-transducer realizing it by the forward construction with C(F3). A1 = 0 1 1 2 3 2 4 5 3 6 7 4 2 5 5 6 8 6 5 6 7 8 3 8 8 5 TM1 = 0 1 2 im E 0 : L 10 : A 11 : B C2 L 00 : L 010 : A ε : O C2 A 00 : A 01 : B 1 : A C2 B 0 : B 10 : A 11 : B…
Figure 9
Figure 9. Figure 9: The core of M1 and a minimal (3, 2)-transducer realizing it by the forward construction with C(F3). The subgroup M2 is obtained from M1 by reflection. 74 [PITH_FULL_IMAGE:figures/full_fig_p074_9.png]
Figure 10
Figure 10. Figure 10: The local operation at a principal state whose transducer coordinate is [PITH_FULL_IMAGE:figures/full_fig_p077_10.png]
Figure 11
Figure 11. Figure 11: The contracted raw product G raw φ (A0). Edge labels are output words. Thus the geometric automaton has states ρ, b bℓ, r, b c, P, D. b Its ordered pairs of children are ρb ( bℓ, rb) bℓ ( bℓ, cb) rb (P, rb) cb (P, cb) P (D, cb) D (P, P). These ordered pairs are all di…
Figure 12
Figure 12. Figure 12: The geometric automaton obtained from G raw φ (A0) after subdivision and folding common initial subpaths. This is A1. Constructing C(H2). We now repeat the same construction with input automaton C(H1) =∼ A1. To distinguish the old a-states of the input automaton from …
Figure 13
Figure 13. Figure 13: The contracted raw product G raw φ (A1). Edge labels are output words. The loops labelled 01 will share their initial 0-edges with the edges labelled 00 after subdivision. The remaining edge from P0 is already one-letter: P0 1 −−→ c. b Similarly, at P1, the two word-l…
Figure 14
Figure 14. Figure 14: The local subdivision step in the construction of [PITH_FULL_IMAGE:figures/full_fig_p083_14.png]
Figure 15
Figure 15. Figure 15: The geometric automaton obtained from G raw φ (A1) after subdividing the word￾labelled edges and folding common initial subpaths. The small labels record the final renaming P0 = a0, D0 = a1, P1 = a2, D1 = a3. 84 [PITH_FULL_IMAGE:figures/full_fig_p084_15.png]
Figure 16
Figure 16. Figure 16: The binary tree with branch set {00, 01, 1} inflates, for n = 3, to the ternary tree with branch set {00, 01, 02, 1, 2}. where u is a father vertex of the original binary tree. Since ub ∈ {0, m} ∗ , we have σn(ub) = 0. Thus old binary leaves satisfy σn(pb) = 0, while …

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

  1. 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.

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