Automorphisms of the generalised Thompson's group $T_{n,r}$

Feyishayo Olukoya

REVIEW 3 major objections 5 minor 20 references

Automorphisms of the generalised Thompson's group $T_{n,r}$

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Pith's one-line read The paper proves that the automorphism group of every generalized Thompson group $T_{n,r}$ consists precisely of the finite, bi-synchronizing transducers that preserve the cyclic order on the circle, and derives the structure of the outer…

desk verdict Completes the automorphism program for T_{n,r}, but the main proof hinges on an unproven proposition in the unpublished BCMNO preprint. read the letter →

arxiv 1908.03816 v1 pith:7NOALU3H submitted 2019-08-10 math.GR

Feyishayo Olukoya This is my paper · ORCID

classification math.GR MSC 20E3620F6520E08

keywords ThompsongroupsHigman-Thompsonautomorphismouterbi-synchronizingtransducersrationalgroupR-infinitypropertytwistedconjugacy

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 sets out to describe, for every $n$ and $r$, the automorphism group of the generalized Thompson group $T_{n,r}$, the circle analogue of the Higman-Thompson groups $G_{n,r}$. Its central theorem identifies $\mathrm{Aut}(T_{n,r})$ with a concrete subgroup of the rational group: the elements that admit a finite initial transducer which is bi-synchronizing, meaning that both the transducer and its inverse reach a fixed state after reading a long enough prefix, and whose induced homeomorphism of Cantor space preserves the cyclic-order relation $\simeq$ that collapses Cantor space to the circle. Because every such transducer is also bi-synchronizing in the sense used for $G_{n,r}$, this shows $\mathrm{Aut}(T_{n,r})$ is a subgroup of $\mathrm{Aut}(G_{n,r})$. From this description the paper derives the structure of the outer automorphism groups: they embed into a single group $\mathrm{Out}(T_{n,n-1})$, form a normal lattice indexed by divisibility in $\mathbb{Z}_{n-1}$, are infinite for $n>2$, and each contains a copy of Thompson's group $F$. The same machinery yields a new infinite family of groups with the $R_\infty$ property: every automorphism of $T_{n,r}$ has infinitely many twisted conjugacy classes.

What carries the argument

The workhorse is the finite initial transducer over the alphabet $X_n$: a finite automaton with a distinguished initial state that reads one letter from the finite set $\dot{r}$ and then processes $n$-adic words, emitting $n$-adic words. The load-bearing property is bi-synchronicity: after a fixed-length prefix, the active state of the transducer, and of its inverse, is forced regardless of the starting state. The paper also uses the reduced node distance between incomparable words to encode the cyclic combinatorics of $T_{n,r}$, and the reduced signature of a core transducer, an element of $\mathbb{Z}_{n-1}$, to control membership in the groups $T O_{n,r}$; the congruence $r\cdot\mathrm{sig} \equiv r \pmod{n-1}$ organizes the divisibility lattice of outer automorphism groups.

What would settle it

Construct a homeomorphism of $C_{n,r}$ that preserves the cyclic-order relation $\simeq$, lies in the rational group, and has infinitely many distinct local actions $h_\nu$; if it normalizes $T_{n,r}$, then the paper's finite-local-action corollary fails and Theorem 5.4 cannot hold. The paper's machinery predicts instead that every normalizer element has finitely many local-action types, so this is a direct falsifier.

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Extended reading notes

Core claim

The central discovery is an exact transducer description of the automorphisms of $T_{n,r}$. Working inside the rational group $R_{n,r}$ of homeomorphisms of the Cantor space $C_{n,r}$ generated by finite initial transducers over the alphabet $X_n$, the paper proves that $\mathrm{Aut}(T_{n,r})$ is isomorphic to the subgroup $T B_{n,r}$ consisting of those elements of $R_{n,r}$ that can be represented by a finite, initial, bi-synchronizing transducer and whose induced map preserves the equivalence relation $\simeq$ identifying Cantor space with the circle $S_r$. The proof adapts the corresponding argument for $G_{n,r}$: a normalizer element of $T_{n,r}$ must preserve the tail equivalence relation $\sim_t$, act on incomparable cones almost in the same fashion, and therefore have only finitely many local actions; these facts force the minimal transducer to be finite and bi-synchronizing, and preservation of $\simeq$ is shown to be necessary and sufficient. A direct corollary is the strict inclusion $\mathrm{Aut}(T_{n,r}) < \mathrm{Aut}(G_{n,r})$.

Load-bearing premise

The whole transducer description rests on the imported proposition that a homeomorphism of Cantor space whose local actions on $\tau\chi$ and $\eta\chi$ differ for every suffix $\chi$ cannot preserve the tail equivalence relation; if that proposition failed, the bi-synchronizing characterization of $\mathrm{Aut}(T_{n,r})$ would collapse.

Editorial extensions

If this is right

Every automorphism of $T_{n,r}$ is an automorphism of $G_{n,r}$, so $\mathrm{Aut}(T_{n,r})$ is a subgroup of $\mathrm{Aut}(G_{n,r})$.
The groups $\mathrm{Out}(T_{n,r})$ embed in $\mathrm{Out}(T_{n,n-1})$; divisibility in the cyclic group $\mathbb{Z}_{n-1}$ gives containments $T O_{n,i} \leq T O_{n,j}$, with $T O_{n,1}$ contained in every $T O_{n,r}$.
For $n>2$, every $\mathrm{Out}(T_{n,r})$ is infinite and contains a copy of Thompson's group $F$.
$\mathrm{Out}(T_{n,r}) = \mathrm{Out}(T_{n,d})$ whenever $d = \gcd(n-1,r)$, but the converse isomorphism question has a negative answer (the case $n=7$), so equal gcd is not necessary for isomorphic outer automorphism groups.
Every $T_{n,r}$ has the $R_\infty$ property: for every automorphism, the twisted conjugacy relation has infinitely many equivalence classes.

Reading between the lines

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

Because both conditions in the characterization are properties of a finite transducer, membership in $\mathrm{Aut}(T_{n,r})$ and in $\mathrm{Out}(T_{n,r})$ is decidable for a given element of the rational group; the paper's algorithms for $O_n$ transfer to $T O_n$.
If the signature homomorphism is surjective on $T O_n$, the join of $\mathrm{Out}(T_{n,r})$ and $\mathrm{Out}(T_{n,s})$ is exactly $\mathrm{Out}(T_{n,\mathrm{lcm}(r,s)})$; surjectivity is proved for $O_n$ whenever divisors of $n$ generate the units of $\mathbb{Z}_{n-1}$, leaving the $T O_n$ case open.
A natural next step suggested by the methods is an automorphism description for the $F$-analogues of these groups, obtained by running the same local-action and synchronizing argument in the $F$ setting.
The $R_\infty$ property gives every automorphism of $T_{n,r}$ an infinite Reidemeister number, placing these groups alongside the known examples in the twisted-conjugacy literature.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here is what you should know about arXiv:1908.03816. It is a substantive extension of BCMNO's transducer-based automorphism program to the T-family of Higman-Thompson groups. The headline results are real: Aut(T_{n,r}) is characterized as the bi-synchronizing transducers preserving the cyclic order, Out(T_{n,r}) embeds into Out(T_{n,n-1}), all Out(T_{n,r}) for n>2 are infinite and contain a copy of Thompson's group F, and the paper answers BCMNO's gcd question negatively and proves the R-infinity property for T_{n,r}. If correct, this is a major within-field advance, and it goes cleanly beyond Brin-Guzman, whose methods only treated r = n-1.

Credit where due: the paper is careful and detailed, and the adaptation to T is not a routine translation. The transitivity arguments for T are genuinely different from those for G, and the lattice of subgroups Out(X_{n,r}) plus the signature homomorphism are presented surprisingly cleanly. The concrete transducer computations (like the element of TO4 that lies in no TO4,r for r<3) are explicit and checkable.

Soft spots: the main route to finite local actions passes through Proposition 4.13 in the unpublished BCMNO preprint [3]. That proposition is stated without proof, and the stress-test note is right: if it fails, Corollary 4.16 and the finite-transducer characterization collapse. This is a genuine verifiability risk, not an internal inconsistency. The paper needs either a proof of Proposition 4.13 or a public version of [3] that establishes it. There are also two statement-level errors: Theorem 10.2 says "T_{n,r} is infinite" where Out(T_{n,r}) was intended, and Theorem 11.8's range "1 ≤ r < n-1" does not match the proof, which treats r=1 and then all r>1; the intended range needs a careful fix. The heavy reliance on [3] is understandable given the program, but it makes this paper hostage to that preprint's status.

Who is this for? Specialists in Higman-Thompson groups, automorphism groups of homeomorphism groups, and transducer methods. This is not a paradigm shift, but it is a serious within-field result. I would send it to a competent referee and let them push on the dependence on [3]. My own verdict is conditional: fix the reliance on [3], correct the statement errors, and this paper is publishable.

Referee Report

3 major / 5 minor

Summary. The paper characterizes the automorphism group of the generalized Thompson group T_{n,r} as the subgroup of the rational group R_{n,r} consisting of elements represented by finite, initial, bi-synchronizing transducers that preserve the cyclic order relation ≃ on the Cantor space (Theorem 5.4). It defines a corresponding subgroup T O_{n,r} of the transducer group O_{n,r} from Bleak–Cameron–Maissel–Navas–Olukoya (BCMNO), and proves that Out(T_{n,r}) is isomorphic to T O_{n,r}. The paper then establishes the nesting structure T O_{n,1} ⊴ T O_{n,r} ⊴ T O_{n,n-1}, equality T O_{n,r} = T O_{n,gcd(n-1,r)}, a negative answer to the BCMNO question on isomorphism of Out(G_{n,r}) versus gcd, the existence of a copy of Thompson's group F in Out(T_{n,r}) for n ≥ 3, and the R∞ property for all T_{n,r}. The proofs use transducer dynamics, viable combinations, and explicit transducer constructions.

Significance. Assuming the validity of the imported Proposition 4.13 from [3], the main theorem is a natural and substantial extension of the BCMNO characterization of Aut(G_{n,r}) to the T_{n,r} family. The structural results for Out(T_{n,r}) (infinite, containing F, lattice relations, R∞ property) were previously known only for special cases, and the negative answer to the gcd question is a concrete new contribution. The paper is detailed and provides explicit transducers for key examples, including the element of T O_4 that is not in any T O_{4,r} for r < 3, and the generators of the F-subgroup. The main results are coherent and mostly self-contained except for the reliance on [3]. If Proposition 4.13 is correct, the paper meets the standard for acceptance; the remaining issues are local corrections.

major comments (3)

[§4, Proposition 4.13] Proposition 4.13 is stated without proof and is attributed to the unpublished preprint [3] (arXiv:1605.09302). This proposition is load-bearing: its contrapositive is used in Corollary 4.14 to produce, for each residue class i ∈ {0,1,...,n-2}, a pair (τ,η) with h_τ = h_η; Lemma 4.15 then uses this to show that h acts 'almost in the same fashion' on any two disjoint cones, and Corollary 4.16 derives the finiteness of local action types. These results are the basis for Lemma 5.1 and hence for Theorems 5.3 and 5.4, the paper's main characterization. Since the manuscript does not prove Proposition 4.13 and [3] is an unpublished preprint, the central claim is not fully verifiable from the manuscript alone. Please provide a proof of Proposition 4.13, or a precise reference to a published version, or at least state the location in [3] where the proof appears and confirm its status.
[§10, Theorem 10.2] The statement 'For n > 3 and 1 ≤ r ≤ n − 1, the group T_{n,r} is infinite' is trivially true, since T_{n,r} is an infinite group. The intended statement, as the surrounding text indicates, is that Out(T_{n,r}) is infinite. Moreover, the proof, which uses the transducers in Figures 2 and 3, requires only n > 2. Please correct the theorem to: 'For n > 2 and 1 ≤ r ≤ n − 1, the group Out(T_{n,r}) is infinite.'
[§11, Theorem 11.8] The theorem states 'The group T_{n,r} for 1 ≤ r < n − 1 has the R∞ property', but the abstract and the introduction claim the R∞ property for all T_{n,r}. The proof in Section 11 appears to cover all 1 ≤ r ≤ n − 1: the orientation-preserving case treats r > 1 explicitly, and the orientation-reversing case treats 1 < r < n. Please correct the range in the theorem statement to '1 ≤ r ≤ n − 1' (with the case n=2, r=1 handled by [7]) and make the abstract and theorem consistent.

minor comments (5)

[Abstract and Theorem 1.1] The phrase 'the induced homeomorphisms on Cantor space respects the cyclic ordering' should read 'respect the cyclic ordering' (subject–verb agreement). The same wording appears in Theorem 5.4.
[Corollary 1.4] The statement 'For 1 ≤ 1 ≤ n − 1' contains a typo; it should be 'For 1 ≤ i ≤ n − 1'.
[Throughout, e.g., §4, Proposition 4.13] The word 'Proposition' is consistently misspelled as 'Propostion' in the manuscript (e.g., Proposition 4.13, Proposition 7.7). Please correct all occurrences.
[Theorem 1.10] The phrase 'is unto the group of units' should be 'is onto the group of units'.
[§7, Theorem 1.9 and Remark 7.19] Theorem 1.9 states the existence of a homomorphism from X_n to the group of units of Z_{n−1} with kernel X_{n,1} for X = T and G. The proof in §7 gives the construction for O_n (Theorem 7.15) and Remark 7.19 says the results carry over to T O_n without further detail. Since the theorem is stated for both, please add an explicit statement (or one sentence in Remark 7.19) that sig|_{T O_n} is a homomorphism with kernel T O_{n,1}, using Proposition 7.7.

Circularity Check

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No circularity: Aut(T_{n,r}) characterization is derived from new transitivity lemmas plus independent BCMNO machinery for G_{n,r}; heavy reliance on [3] is a dependency, not a reduction.

full rationale

The central theorem (Theorem 5.4) is not assumed in the paper's inputs. The proof derives it from the new transitivity results for T_{n,r} (Lemmas 2.1 and 3.4), the normalizer reduction to cyclic-order-preserving homeomorphisms (Lemmas 3.5 and 3.6), and the finite-local-action argument in Section 4. That argument imports Proposition 4.13 from BCMNO [3] to obtain Corollary 4.14, which feeds Lemma 4.15 and Corollary 4.16; these are then used to prove the bi-synchronizing transducer representation in Theorem 5.3. The dependence on [3] is indeed load-bearing, and [3] is a preprint coauthored by the present author, so the paper is not fully self-contained. However, this is reliance on an external (though overlapping-author) result about the Higman groups G_{n,r} and tail equivalence, not an equation-level reduction: no conclusion of the present paper is used as a hypothesis, and the new claims—the Aut(T_{n,r}) characterization, the containment/lattice results, the negative answer to the gcd question, and the R_infinity property—are not fitted parameters, renamed known results, or definitions of the target in terms of itself. The reader-flagged risk that Corollary 4.16 collapses if Proposition 4.13 is false is a verifiability and correctness concern, located at Proposition 4.13 / Corollaries 4.14 and 4.16, but it is not circularity under the stated hard rules.

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

Everything load-bearing is mathematical: Rubin's theorem, the BCMNO transducer framework for G_{n,r} (including viable combinations and the membership criterion), Grigorchuk-Nekrashevych-Suschanski minimalization, and Brin-Guzman's results for T_{n,n-1}. No numerical parameters are fitted and no new entities are postulated. The most fragile imported ingredient is Proposition 4.13 of [3], on which the finite-local-actions proof rests.

assumptions (4)

standard math Rubin's theorem: for sufficiently rich groups of homeomorphisms of a compact space, the automorphism group is isomorphic to the normalizer in the homeomorphism group.
Used in Theorem 3.2 to identify Aut(T_{n,r}) with N_{H(S_r)}(T_{n,r}).
domain assumption BCMNO characterization of Aut(G_{n,r}) as the group B_{n,r} of bi-synchronizing transducers, and the associated group O_n with the viable combination membership criterion (Lemma 6.9).
The paper imports the entire transducer framework and membership criterion from [3] (a preprint by the author and collaborators) and uses it as a black box in Sections 4-7.
domain assumption Grigorchuk-Nekrashevych-Suschanski transducer minimalization theory, including the existence of a unique minimal transducer under omega-equivalence and the inverse transducer construction.
Used throughout Section 4 to represent homeomorphisms by finite minimal transducers.
domain assumption Brin-Guzman results on Out(T_{n,n-1}) being infinite and containing a copy of Thompson's group F for n ≥ 3.
Baseline extended in Theorem 10.7; their methods for automorphisms of F_n and T_{n,n-1} inform Section 3.

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Pith. "Pith review of Automorphisms of the generalised Thompson's group $T_{n,r}$." pith.science (2026). https://pith.science/paper/7NOALU3H

@misc{pith2026190803816,
  author       = {Pith},
  title        = {Pith review of: Automorphisms of the generalised Thompson's group $T_n,r$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7NOALU3H}},
  note         = {Machine review of arXiv:1908.03816}
}

abstract

The recent paper "The further chameleon groups of Richard Thompson and Graham Higman: automorphisms via dynamics for the Higman groups $G_{n,r}$" of Bleak, Cameron, Maissel, Navas and Olukoya (BCMNO) characterises the automorphisms of the Higman-Thompson groups $G_{n,r}$ as the specific subgroup of the rational group $\mathcal{R}_{n,r}$ of Grigorchuk, Nekrashevych and Suchanski{\u i}'s consisting of those elements which have the additional property of being bi-synchronizing. In this article, we extend the arguments of BCMNO and characterise the automorphism group of $T_{n,r}$ as a subgroup of $\mathrm{Aut}{G_{n,r}}$. We then show that the groups $\mathrm{Out}{T_{n,r}}$ can be identified with subgroups of the group $\mathrm{Out}{T_{n,n-1}}$. Extending results of Brin and Guzman, we show that the groups $\mathrm{Out}{T_{n,r}}$, for $n>2$, are all infinite and contain an isomorphic copy of Thompson's group $F$. For $X \in \{T,G\}$, we study the groups $\mathrm{Out}{X_{n,r}}$ and show that these fit in a lattice structure where $\mathrm{Out}{X_{n,1}} \unlhd \mathrm{Out}{X_{n,r}}$ for all $1 \le r \le n-1$ and $\mathrm{Out}{X_{n,r}} \unlhd \mathrm{Out}{X_{n,n-1}}$. This gives a partial answer to a question in BCMNO concerning the normal subgroup structure of $\mathrm{Out}{G_{n,n-1}}$. Furthermore, we deduce that for $1\le j,d \le n-1$ such that $d = \gcd(j, n-1)$, $\mathrm{Out}{X_{n,j}} = \mathrm{Out}{X_{n,d}}$ extending a result of BCMNO for the groups $G_{n,r}$ to the groups $T_{n,r}$. We give a negative answer to the question in BCMNO which asks whether or not $\mathrm{Out}{G_{n,r}} \cong \mathrm{Out}{G_{n,s}}$ if and only if $\gcd(n-1,r) = \gcd(n-1,s)$. We conclude by showing that the groups $T_{n,r}$ have the $R_{\infty}$ property extending the result of Burillo, Matucci and Ventura and, independently, Gon{\c c}alves and Sankaran, for Thompson's group $T$.

Figures

Figures reproduced from arXiv: 1908.03816 by the authors.

**Figure 1.** An element g ∈ T O4 which is not in T O4,r for any 0 ≤ r < 3 . We make the following observations about g which the reader may verify: (i) For all states α of g the map gα : Cn → Cn preserves the lexicographic ordering on Cn. Thus, since g 2 = id under the product defined for T On, by Theorem 6.16 there is some non-zero r ∈ Z4\ such that g ∈ T On,r. (ii) We have im(a) ∩ im(b) = ∅ and im(a) ⊔ im(b) = C4. In particula… view at source ↗

**Figure 2.** An element T ∈ T On,1 of infinite order. The states a and b of T are homeomorphism states, and so T is in fact an element of T On,1. In order to show that T has infinite order we make use of the action of the group On, as introduced in the paper [3], on the space XZ n / hσni where σn is the shift map on XZ n . Let U ∈ On and suppose that k ∈ N is the minimal synchronizing level of U. The action of U on XZ n / hσni i… view at source ↗

**Figure 3.** An element U ∈ T On,1 of infinite order. The state p of U is a homeomorphism state and moreover U is an element of T On,1 of infinite order. In order to state our next result we require the following notion and result from [3]. Definition 10.3. Given an element g = hXn, Qg, πg, λgi ∈ On a state q of g is called a loop state if there is some i ∈ Xn such that πg(i, q) = q. The following lemma is proven in [3]: Lemma 1… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: The subtransducers A and B of T and U respectively. Notice that Aa and Bp induce self-homeomorphisms of the Cantor space {0, n − 1} ω equal, respectively, to the restrictions of the homeomorphisms Ta and Up to the space {0, n−1} ω . It is not hard to see that F ∼= hAa,…

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url: http://www.numdam.org/item?id=PMIHES_1996__84__5_0

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Reviewed August 14, 2026 · model on record in the stance chip above.

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The further chameleon groups of Richard Thompson and Graham Higman: Automorphisms via dynamics for the Higman groups $G_{n,r}$

Collin Bleak, Peter Cameron, Yonah Maissel, Andr´ es Nav as, and Feyishayo Olukoya. The furhter chameleon groups of Richard Thompson and Graham Higm an: Automorphisms via dynamics for the Higman groups Gn,r. Version 2. 2019. ArXiv: 1605.09302 (math.GR)

work page Pith review arXiv 2019

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T he conjugacy problem in extensions of Thompson’s group F

Jos´ e Burillo, Francesco Matucci, and Enric Ventura. “T he conjugacy problem in extensions of Thompson’s group F ”. In: Israel J. Math. 216.1 (2016), pp. 15–59. issn: 0021-2172. doi: 10.1007/s11856-016-1403-9 . url: https://doi.org/10.1007/s11856-016-1403-9

work page doi:10.1007/s11856-016-1403-9 2016

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On De Bruijn automorphisms of shift spaces

Collin Bleak, Peter Cameron, and Feyishayo Olukoya. On De Bruijn automorphisms of shift spaces. In Preparation, 2019

work page 2019

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Twisted conjugacy classes in R. Thompson’s group F

Collin Bleak, Alexander Fel ′shtyn, and Daciberg L. Gon¸ calves. “Twisted conjugacy classes in R. Thompson’s group F ”. In: Paciﬁc J. Math. 238.1 (2008), pp. 1–6. issn: 0030-8730. doi: 10.2140/pjm.2008.238.1. url: http://dx.doi.org/10.2140/pjm.2008.238.1

work page doi:10.2140/pjm.2008.238.1 2008

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The chameleon groups of Richard J. Thom pson: automorphisms and dynamics

Matthew G. Brin. “The chameleon groups of Richard J. Thom pson: automorphisms and dynamics”. In: Inst. Hautes ´Etudes Sci. Publ. Math. 84 (1996), 5–33 (1997). issn: 0073-

work page 1996

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Automorphisms of generalized Thompson groups

Matthew G. Brin and Fernando Guzm´ an. “Automorphisms of generalized Thompson groups”. In: J. Algebra 203.1 (1998), pp. 285–348. issn: 0021-8693. doi: 10.1006/jabr.1997.7315. url: http://dx.doi.org/10.1006/jabr.1997.7315. 42

work page arXiv 1998

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The automorphism group o f Thompson’s group F : subgroups and metric properties

Jos´ e Burillo and Sean Cleary. “The automorphism group o f Thompson’s group F : subgroups and metric properties”. In: Rev. Mat. Iberoam. 29.3 (2013), pp. 809–828. issn: 0213-2230. doi: 10.4171/RMI/741. url: https://doi.org/10.4171/RMI/741

doi:10.4171/rmi/741 2013

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Isomorphisms of Brin-Higman-Thompson groups

Warren Dicks and Conchita Mart ´ ınez-P´ erez. “Isomorphisms of Brin-Higman-Thompson groups”. In: Israel Journal of Mathematics 199.1 (2014), pp. 189–218. issn: 1565-8511. doi: 10.1007/s11856-013-0042-7 . url: https://doi.org/10.1007/s11856-013-0042-7

work page doi:10.1007/s11856-013-0042-7 2014