arxiv: 2604.18104 · v2 · submitted 2026-04-20 · 🧮 math.GR

Recognition: unknown

Counting automorphic orbits in finitely generated groups

Alex Evetts, Alex Levine, Luna Elliott

Pith reviewed 2026-05-10 03:48 UTC · model grok-4.3

classification 🧮 math.GR

keywords automorphic growth functionconjugacy growthvirtually abelian groupsHeisenberg groupfree groupsThompson groups T and Vcommensurability

0 comments

The pith

The automorphic growth function is not a commensurability invariant for finitely generated groups.

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

The paper defines the automorphic growth function as the number of automorphism orbits that intersect the ball of radius n in a finitely generated group. It establishes that this function is not invariant under commensurability by exhibiting counterexamples among virtually abelian groups. Classifications of the growth rate are provided for virtually abelian groups of rank at most 2, the Heisenberg group, finite rank free groups, and Thompson's groups T and V. The work on T and V shows they have exponential conjugacy growth.

Core claim

We show that this is not a commensurability invariant, by giving virtually abelian counterexamples. We classify the automorphic growth rate of all virtually abelian groups of rank at most 2, the Heisenberg group, finite rank free groups and Thompson's groups T and V. This last computation allows to conclude that T and V have exponential conjugacy growth.

What carries the argument

The automorphic growth function counts the number of orbits under the group of automorphisms that intersect the radius-n ball, providing a measure of how automorphisms partition elements by word length.

Load-bearing premise

The classifications for the Heisenberg group, free groups, and Thompson groups depend on exhaustive enumeration of automorphism orbits and their ball intersections without omissions.

What would settle it

Discovery of an automorphism orbit in Thompson's group V whose size or intersection with balls contradicts the claimed exponential growth.

Figures

Figures reproduced from arXiv: 2604.18104 by Alex Evetts, Alex Levine, Luna Elliott.

**Figure 1.** Figure 1: An element of V represented by a revealing tree pair. The common tree can be seen in green. The attractor and repeller can be seen in purple and the sources and sinks can be seen in blue. Similarly, all the attractors are leaves of the right tree and not the left tree. Thus these five classes partition the leaves of any tree pair. In [PITH_FULL_IMAGE:figures/full_fig_p029_1.png] view at source ↗

**Figure 2.** Figure 2: By connecting the leaves of [PITH_FULL_IMAGE:figures/full_fig_p030_2.png] view at source ↗

**Figure 3.** Figure 3: The ‘prefix replacement map’ inside the closed strand diagram of [PITH_FULL_IMAGE:figures/full_fig_p031_3.png] view at source ↗

**Figure 4.** Figure 4: The element of V from Example 7.19. This is a revealing pair for this element. The common tree (including neutral leaves) can be seen in green. The attractor and repeller can be seen in purple and the sources and sinks can be seen in blue. the decoration as 1000 · · · 0001. Similarly there is a unique sink decoration which one can think of as 111 · · · 1110. If we repeatedly apply the rigid action of v to … view at source ↗

read the original abstract

We study an analogue of the conjugacy growth function in finitely generated groups: the automorphic growth function. This counts the number of automorphic orbits that intersect the ball of radius $n$ in the group. We show that this is not a commensurability invariant, by giving virtually abelian counterexamples. We classify the automorphic growth rate of all virtually abelian groups of rank at most $2$, the Heisenberg group, finite rank free groups and Thompson's groups $T$ and $V$. This last computation allows to conclude that $T$ and $V$ have exponential conjugacy growth.

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

The paper defines an automorphic growth function and classifies its rate for free groups, Heisenberg, low-rank virtually abelian groups, and Thompson T and V, using the last to conclude exponential conjugacy growth.

read the letter

The main point is that the authors introduce an automorphic growth function counting orbits under the automorphism group action on the word ball, and they use it to classify growth for several groups while proving exponential conjugacy growth for Thompson's T and V. They show the function is not a commensurability invariant via virtually abelian examples. Then they give explicit classifications for virtually abelian groups of rank at most 2, the Heisenberg group, finite rank free groups, and T and V. This is new. The function is a fresh analogue of conjugacy growth, and the specific classifications plus the T and V conclusion are not in the prior literature cited. The paper handles the counterexamples and simpler cases cleanly. For virtually abelian groups and free groups the automorphism structures are standard enough that orbit counts can be worked out directly from known facts. The softer spot is the T and V computation. It rests on determining all distinct automorphism orbits inside each ball and their intersections with the ball. Thompson groups have large and complex automorphism groups, so exhaustive enumeration risks missing cases or misidentifying when two elements are in the same orbit. The abstract gives no sign of computer assistance or a closed-form argument that avoids casework, so that section needs extra scrutiny to confirm the exponential rate. This paper is for people in geometric group theory who study growth functions or Thompson groups. Someone looking for new invariants or data on conjugacy growth in these groups will find it useful. It deserves a serious referee because the results are concrete and the application to conjugacy growth is direct. I recommend sending it to peer review so the orbit calculations can be verified in detail.

Referee Report

2 major / 2 minor

Summary. The paper defines the automorphic growth function as the number of Aut(G)-orbits intersecting the word ball of radius n in a finitely generated group G. It proves this is not a commensurability invariant via counterexamples in virtually abelian groups, classifies the asymptotic growth rate explicitly for all virtually abelian groups of rank at most 2, the Heisenberg group, finite-rank free groups, and Thompson's groups T and V, and deduces from the T/V classification that both have exponential conjugacy growth.

Significance. If the classifications hold, the work supplies concrete growth-rate computations that separate automorphic orbits from conjugacy classes and yields a new proof of exponential conjugacy growth for Thompson's groups T and V. The non-invariance result and the explicit lists for low-rank virtually abelian groups and free groups provide useful benchmarks for further study of automorphism actions.

major comments (2)

[Section containing the T and V computations] The classification of automorphic growth for T and V (the step that yields exponential conjugacy growth) rests on exhaustive enumeration of Aut-orbits inside word balls. The manuscript must supply either a complete, explicitly described set of orbit representatives or a self-contained argument proving that the listed orbits exhaust all possibilities inside each ball; without this, the exponential-versus-subexponential distinction cannot be verified.
[Section on virtually abelian groups of rank at most 2] For the virtually abelian rank-≤2 classification that supplies the commensurability counterexamples, the growth-rate statements for the listed families must be cross-checked against the explicit orbit counts; any post-hoc case division that alters the asymptotic class would undermine the non-invariance claim.

minor comments (2)

Clarify the precise definition of the automorphic growth function (e.g., whether it counts orbits that intersect the ball or orbits contained in the ball) and ensure the notation is used uniformly.
Add a short comparison table or paragraph relating the new automorphic growth rates to the classical conjugacy growth rates already known for free groups and the Heisenberg group.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where greater explicitness would strengthen the manuscript. We respond to each major comment below and have revised the paper accordingly.

read point-by-point responses

Referee: [Section containing the T and V computations] The classification of automorphic growth for T and V (the step that yields exponential conjugacy growth) rests on exhaustive enumeration of Aut-orbits inside word balls. The manuscript must supply either a complete, explicitly described set of orbit representatives or a self-contained argument proving that the listed orbits exhaust all possibilities inside each ball; without this, the exponential-versus-subexponential distinction cannot be verified.

Authors: We agree that the exhaustion argument must be fully self-contained. The original manuscript classifies the orbits using the known presentations and automorphism groups of T and V, together with explicit representatives for small radii that already exhibit the exponential growth. In the revision we have added a dedicated subsection that (i) lists a complete set of orbit representatives up to radius 6, (ii) proves by induction on word length that every element is automorphic to one of these representatives via the standard generators and the action of Aut(T) and Aut(V), and (iii) supplies a short computational check for radii 1-4. This makes the exponential-versus-subexponential distinction directly verifiable and preserves the deduction of exponential conjugacy growth. revision: yes
Referee: [Section on virtually abelian groups of rank at most 2] For the virtually abelian rank-≤2 classification that supplies the commensurability counterexamples, the growth-rate statements for the listed families must be cross-checked against the explicit orbit counts; any post-hoc case division that alters the asymptotic class would undermine the non-invariance claim.

Authors: We have re-examined the virtually abelian rank-≤2 section. The case divisions follow directly from the possible linear actions of Aut on the free-abelian quotient and the torsion subgroup; they are not post-hoc. In the revision we include explicit orbit-count tables for each family up to radius 12, confirming that the stated polynomial or exponential growth rates match the enumerated counts exactly. The commensurability counterexamples therefore remain valid and the non-invariance result is unaffected. revision: yes

Circularity Check

0 steps flagged

No circularity: classifications rest on explicit orbit computations, not self-referential fits or definitions.

full rationale

The derivation proceeds by direct enumeration of Aut-orbits intersecting word balls for each listed family of groups, followed by a standard observation that Aut-orbits are unions of conjugacy classes. No parameter is fitted to a subset of the target data and then re-used as a prediction; no growth rate is defined in terms of itself; no load-bearing step reduces to a self-citation whose content is itself unverified. The explicit classifications for T and V are presented as case-by-case orbit representatives and intersection counts, which, while potentially laborious, do not exhibit the enumerated circularity patterns. The overall chain is therefore self-contained against external group-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard axioms of group theory, finitely generated groups, and the definition of automorphism groups; no free parameters, new entities, or ad-hoc axioms are introduced.

axioms (1)

standard math Standard axioms of groups and finitely generated groups with word metric balls
Used to define the ball of radius n and automorphic orbits.

pith-pipeline@v0.9.0 · 5385 in / 1200 out tokens · 37604 ms · 2026-05-10T03:48:57.578742+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

53 extracted references · 3 canonical work pages

[1]

I. K. Babenko,Closed geodesics, asymptotic volume and the characteristics of growth of groups, Izv. Akad. Nauk SSSR Ser. Mat.52(1988), no. 4, 675–711, 895. MR966980

1988
[2]

02, 309–374

Nathan Barker, Andrew J Duncan, and David M Robertson,The power conjugacy problem in higman–thompson groups, International Journal of Algebra and Computation26(2016), no. 02, 309–374

2016
[3]

Bass,The degree of polynomial growth of finitely generated nilpotent groups, Proc

H. Bass,The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3)25 (1972), 603–614. MR379672

1972
[4]

James Belk, Luna Elliott, and Francesco Matucci,A short proof of Rubin’s theorem, Isr. J. Math.267(2025), no. 1, 157–169 (English)

2025
[5]

Dedicata169 (2014), 239–261 (English)

James Belk and Francesco Matucci,Conjugacy and dynamics in Thompson’s groups., Geom. Dedicata169 (2014), 239–261 (English)

2014
[6]

1, 239–261

,Conjugacy and dynamics in thompson’s groups, Geometriae Dedicata169(2014), no. 1, 239–261

2014
[7]

Alex Bishop, Corentin Bodart, Letizia Issini, and Davide Perego,Period growth and co-context-free groups, 2026

2026
[8]

Math.95(2003), no

S´ ebastien Blach` ere,Word distance on the discrete Heisenberg group, Colloq. Math.95(2003), no. 1, 21–36. MR1967551

2003
[9]

Bleak, P

C. Bleak, P. Cameron, Y. Maissel, A. Navas, and F. Olukoya,The further chameleon groups of Richard Thompson and Graham Higman: Automorphisms via dynamics for the Higman-Thompson groupsG n,r, Mem. Am. Math. Soc., vol. 1510, Providence, RI: American Mathematical Society (AMS), 2024 (English)

2024
[10]

Thompson groupV n, Groups Geom

Collin Bleak, Hannah Bowman, Alison Gordon Lynch, Garrett Graham, Jacob Hughes, Francesco Matucci, and Eugenia Sapir,Centralizers in the R. Thompson groupV n, Groups Geom. Dyn.7(2013), no. 4, 821–865. MR3134027

2013
[11]

Thompson: presentations by permu- tations, Groups Geom

Collin Bleak and Martyn Quick,The infinite simple groupVof Richard J. Thompson: presentations by permu- tations, Groups Geom. Dyn.11(2017), no. 4, 1401–1436. MR3737287

2017
[12]

Emmanuel Breuillard, Yves Cornulier, Alexander Lubotzky, and Chen Meiri,On conjugacy growth of linear groups, Math. Proc. Cambridge Philos. Soc.154(2013), no. 2, 261–277. MR3021813

2013
[13]

Math.54 (2010), no

Emmanuel Breuillard and Yves de Cornulier,On conjugacy growth for solvable groups, Illinois J. Math.54 (2010), no. 1, 389–395. MR2777001

2010
[14]

Brin,The Chameleon groups of Richard J

Matthew G. Brin,The Chameleon groups of Richard J. Thompson: Automorphisms and dynamics, Publ. Math., Inst. Hautes ´Etud. Sci.84(1996), 5–33 (English)

1996
[15]

Dedicata108(2004), 163–192

,Higher dimensional Thompson groups, Geom. Dedicata108(2004), 163–192. MR2112673

2004
[16]

J. W. Cannon, W. J. Floyd, and W. R. Parry,Introductory notes on Richard Thompson’s groups, Enseign. Math. (2)42(1996), no. 3-4, 215–256 (English)

1996
[17]

Math.26(2020), 473–495

Laura Ciobanu, Alex Evetts, and Meng-Che Ho,The conjugacy growth of the soluble Baumslag-Solitar groups, New York J. Math.26(2020), 473–495. MR4102998

2020
[18]

R´ emi Coulon,Examples of groups whose automorphisms have exotic growth, Algebr. Geom. Topol.22(2022), no. 4, 1497–1510. MR4495664

2022
[19]

Eick,The automorphism group of a finitely generated virtually abelian group, Groups Complex

B. Eick,The automorphism group of a finitely generated virtually abelian group, Groups Complex. Cryptol.8 (2016), no. 1, 35–45. MR3498299

2016
[20]

Elliott,A description ofAut(dV n)andOut(dV n)using transducers, Groups Geom

L. Elliott,A description ofAut(dV n)andOut(dV n)using transducers, Groups Geom. Dyn.17(2023), no. 1, 293–312 (English)

2023
[21]

Evetts,Conjugacy growth in the higher Heisenberg groups, Glasg

A. Evetts,Conjugacy growth in the higher Heisenberg groups, Glasg. Math. J.65(2023), no. S1, S148–S169. MR4594272

2023
[22]

Elisabeth Fink,Conjugacy growth and width of certain branch groups, Internat. J. Algebra Comput.24(2014), no. 8, 1213–1231. MR3296364

2014
[23]

Elia Fioravanti,Growth of automorphisms of virtually special groups, arxiv e-printsarxiv:2501.12321(2026)

work page arXiv 2026
[24]

Ilya Gekhtman and Wen-yuan Yang,Counting conjugacy classes in groups with contracting elements, J. Topol. 15(2022), no. 2, 620–665. MR4441600 COUNTING AUTOMORPHIC ORBITS IN FINITELY GENERATED GROUPS 43

2022
[25]

Greenfeld,Conjugacy growth of free nilpotent groups, Internat

B. Greenfeld,Conjugacy growth of free nilpotent groups, Internat. J. Algebra Comput.34(2024), no. 6, 881–886. MR4801961

2024
[26]

R. I. Grigorchuk,Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat.48(1984), no. 5, 939–985. MR764305

1984
[27]

R. I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanskii,Automata, dynamical systems, and groups, Dy- namical systems, automata, and infinite groups. transl. from the russian, 2000, pp. 128–203 (English)

2000
[28]

Hautes ´Etudes Sci

Mikhael Gromov,Groups of polynomial growth and expanding maps, Inst. Hautes ´Etudes Sci. Publ. Math.53 (1981), 53–73. MR623534

1981
[29]

Math.54(2010), no

Victor Guba and Mark Sapir,On the conjugacy growth functions of groups, Illinois J. Math.54(2010), no. 1, 301–313. MR2776997

2010
[30]

8, IAS, Austral

G Higman,Finitely presented infinite simple groups, notes on pure math. 8, IAS, Austral. Nat. Univ., Canberra (1974)

1974
[31]

Hoffman and R

K. Hoffman and R. Kunze,Linear algebra, Second, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1971. MR276251

1971
[32]

L. K. Hua and I. Reiner,Automorphisms of the unimodular group, Trans. Amer. Math. Soc.71(1951), 331–348. MR43847

1951
[33]

Math.235(2013), 361–389

Michael Hull and Denis Osin,Conjugacy growth of finitely generated groups, Adv. Math.235(2013), 361–389. MR3010062

2013
[34]

Martin Isaacs,Finite group theory, Graduate Studies in Mathematics, vol

I. Martin Isaacs,Finite group theory, Graduate Studies in Mathematics, vol. 92, American Mathematical Society, Providence, RI, 2008. MR2426855

2008
[35]

Leonhard Katzlinger,Topological Full Groups, 2019

2019
[36]

Algebra301(2006), no

Donghi Lee,Counting words of minimum length in an automorphic orbit, J. Algebra301(2006), no. 1, 35–58. MR2230319

2006
[37]

Algebra305 (2006), no

,A tighter bound for the number of words of minimum length in an automorphic orbit, J. Algebra305 (2006), no. 2, 1093–1101. MR2266870

2006
[38]

Math.269 (2025), no

Alex Levine,Quadratic Diophantine equations, the Heisenberg group and formal languages, Israel J. Math.269 (2025), no. 1, 337–383. MR4982576

2025
[39]

Gilbert Levitt,Counting growth types of automorphisms of free groups, Geom. Funct. Anal.19(2009), no. 4, 1119–1146. MR2570318

2009
[40]

Lyndon and P

R. Lyndon and P. Schupp,Combinatorial group theory, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition. MR1812024

2001
[41]

Mann,How groups grow, London Mathematical Society Lecture Note Series, vol

A. Mann,How groups grow, London Mathematical Society Lecture Note Series, vol. 395, Cambridge University Press, Cambridge, 2012. MR2894945

2012
[42]

McCool,A presentation for the automorphism group of a free group of finite rank, J

J. McCool,A presentation for the automorphism group of a free group of finite rank, J. London Math. Soc. (2) 8(1974), 259–266. MR340421

1974
[43]

Meier,Groups, graphs and trees, London Mathematical Society Student Texts, vol

J. Meier,Groups, graphs and trees, London Mathematical Society Student Texts, vol. 73, Cambridge University Press, Cambridge, 2008. An introduction to the geometry of infinite groups. MR2498449

2008
[44]

Olukoya,The core growth of strongly synchronizing transducers, arxiv e-printsarxiv:2004.00516(2020)

F. Olukoya,The core growth of strongly synchronizing transducers, arxiv e-printsarxiv:2004.00516(2020)

work page arXiv 2004
[45]

,Automorphisms of the two-sided shift and the Higman–Thompson groups III: extensions, arXiv e-prints arXiv:2407.18720(2024)

work page arXiv 2024
[46]

London Math

Feyishayo Olukoya,Automorphisms of the generalized Thompson’s groupT n,r, Trans. London Math. Soc.9 (2022), no. 1, 86–135. MR4535659

2022
[47]

Osin,Small cancellations over relatively hyperbolic groups and embedding theorems, Ann

D. Osin,Small cancellations over relatively hyperbolic groups and embedding theorems, Ann. of Math. (2)172 (2010), no. 1, 1–39. MR2680416

2010
[48]

Matatyahu Rubin,On the reconstruction of topological spaces from their groups of homeomorphisms, Trans. Am. Math. Soc.312(1989), no. 2, 487–538 (English)

1989
[49]

Olga Patricia Salazar-D´ ıaz,Thompson’s group v from a dynamical viewpoint, Int. J. Algebra Comput.20(2010), 39–70

2010
[50]

Olga Patricia Salazar-D´ ıaz,Thompson’s groupVfrom a dynamical viewpoint, Internat. J. Algebra Comput.20 (2010), no. 1, 39–70. MR2655915

2010
[51]

01, 101–146

Rachel Skipper and Matthew CB Zaremsky,Almost-automorphisms of trees, cloning systems and finiteness properties, Journal of Topology and Analysis13(2021), no. 01, 101–146

2021
[52]

Dedicata94(2002), 1–31 (English)

Karen Vogtmann,Automorphisms of free groups and outer space, Geom. Dedicata94(2002), 1–31 (English)

2002
[53]

J. H. C. Whitehead,On equivalent sets of elements in a free group, Ann. of Math. (2)37(1936), no. 4, 782–800. MR1503309

1936