Examples of groups whose automorphisms have exotic growth

R\'emi Coulon

Examples of groups whose automorphisms have exotic growth

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Pith's one-line read For any finite collection of computable length functions on the integers, there is a finitely generated group whose outer automorphism makes every nontrivial conjugacy class grow, at logarithmic scale, according to a sum of those length…

desk verdict Solid, self-contained note: Prop 2.1 is the real contribution, the applications are genuine, and the only issues are a typo in Definition 1.2(iv) and an in-preparation background citation. read the letter →

arxiv 1908.11668 v1 pith:A56BCCFQ submitted 2019-08-30 math.GR

Rémi Coulon This is my paper · ORCID

classification math.GR MSC 20E3620F2820F6520F6720F06

keywords growthofautomorphismsouterautomorphismgroupsconjugacyclasseshyperbolicRipsconstructiondistortionsubgroupslengthfunctionsLipschitzmetric

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 establishes that the growth of a conjugacy class under an outer automorphism is not governed by a universal dichotomy: for any finite list of computable length functions on the integers, there is a finitely generated group and an outer automorphism such that each nontrivial conjugacy class grows, at logarithmic scale, like a sum of some of those length functions, and each listed function occurs. This matters because all previously studied classes of groups, including free abelian, free, surface, torsion-free hyperbolic, and torsion-free toric relatively hyperbolic groups, force such growth to be either polynomial or exponential. The construction shows that dichotomy is special to those classes, not a general law. A concrete instance is an explicit group whose conjugacy-class growth under an automorphism behaves like $\sqrt{n}$ after taking logarithms.

What carries the argument

The mechanism is Proposition 2.1, a quasi-isometric embedding of the quotient group $Q$ into $\mathrm{Out}(N)$: for a non-elementary hyperbolic group $G$ with a finitely generated normal subgroup $N$, the word metric on $Q = G/N$ is quasi-isometric to the Lipschitz metric on $\mathrm{Out}(N)$ induced by conjugation. The upper bound is immediate from Lipschitz constants of generators; the lower bound uses Lemma 2.4, the exponential divergence of geodesics in a hyperbolic space. If a conjugacy class of $N$ is pushed by the automorphism associated to $q \in Q$, its length grows at least like $\lambda^{|q|}$, so the logarithm of the norm tracks the word length of $q$. A distortion theorem for finitely presented groups, stated as Theorem 3.2, supplies a group $Q$ containing an element whose word length is equivalent to any given computable length function on $\mathbb{Z}$, and the Rips construction provides the hyperbolic extension; the direct product of such groups realizes finite collections.

What would settle it

Exhibit a computable length function $L$ on $\mathbb{Z}$ that is not strongly equivalent to the word length of any cyclic subgroup of any finitely presented group; Theorem 1.3 would then fail for that $L$. Concretely, one could compute, for the Heisenberg-type example in Remark 3.5, the actual value of $\ln \|\Phi^n(c)\|$ for the distorted element and check whether it is equivalent to $\sqrt{n}$; a deviation, such as linear growth of the logarithm, would locate the point where the logarithmic equivalence breaks.

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Extended reading notes

Core claim

The central claim is Theorem 1.3: given a finite collection $\mathcal{L}$ of computable length functions on $\mathbb{Z}$, there exists a finitely generated group $G$ and an outer automorphism $\Phi \in \mathrm{Out}(G)$ such that for every nontrivial conjugacy class $c$ of $G$, the map $n \mapsto \ln \|\Phi^n(c)\|$ is equivalent to a linear combination of elements of $\mathcal{L}$, and conversely every $L \in \mathcal{L}$ is realized by some $c$. The author states the result at the level of logarithms: the construction does not pinpoint the exact growth of the norm, but bounds it between $A^{-1}\lambda^{L(n)}$ and $A\lambda^{L(n)}$ for some $\lambda>1$ and $A>0$. The group $G$ is built as a direct product of groups attached to single length functions, each arising from a short exact sequence $1 \to N \to G \to Q \to 1$ with $G$ hyperbolic and torsion-free.

Load-bearing premise

The argument leans on the theorem that every computable length function on $\mathbb{Z}$ is strongly equivalent to the word length of a cyclic subgroup in some finitely presented group; if any computable length function fails to be embeddable this way, the corresponding growth type is lost from the construction.

Editorial extensions

If this is right

The polynomial-or-exponential growth dichotomy of Theorem 1.1 fails for general finitely generated groups; in particular, logarithmic growth types such as $n \mapsto \sqrt{n}$ are realized.
Any finite collection of computable length functions on $\mathbb{Z}$ can be realized simultaneously by one automorphism acting on different conjugacy classes of one group.
With a refinement of the Rips construction, the map $Q \to \mathrm{Out}(N)$ can be made an isomorphism, so the Lipschitz metric on $\mathrm{Out}(G)$ can be quasi-isometric to any finitely presented group $Q$ (Theorem 1.4).
The construction can be combined with other Rips-construction refinements: the group $N$ can be chosen linear, residually finite, or with Kazhdan's property (T) (Remark 3.4), so the exotic growth coexists with strong rigidity properties.

Reading between the lines

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

The logarithmic-scale control suggests that the true growth rates, not only their logarithms, may carry additional information; one can test whether the growth is exactly $\exp(\lambda^{L(n)})$ or has subexponential corrections for specific $L$.
Because the lower bound uses only exponential divergence, the same construction should work in any setting where a group with exponential divergence is extended by a finitely generated normal subgroup, for instance CAT(-1) or relatively hyperbolic groups, giving a broader realization theorem.
The explicit Heisenberg example predicts $\ln \|\Phi^n(c)\| \sim \sqrt{n}$; a direct computation of the word length in the Rips group would either confirm the predicted logarithmic growth or reveal a sharper asymptotic, and could be checked for small $n$.
Since there are countably many computable length functions, the theorem shows the set of possible logarithmic growth types for automorphisms of finitely generated groups is as large as the set of computable functions; whether non-computable length functions can appear is left open by the method.

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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's my take on Coulon's note. The new content is the proof that in the Rips construction the map Q → Out(N) is a quasi-isometric embedding with respect to the Lipschitz metric (Prop 2.1). That is a real step: the upper bound is easy, but the lower bound needs exponential divergence of hyperbolic spaces and a careful argument about shortest preimages. I went through the proof and it holds together; the use of the Morse property for cyclic subgroups is legitimate. The applications then fall out: Theorem 1.3 says you can realize any finite set of computable length functions on Z as the log-growth of conjugacy classes under one outer automorphism, and Theorem 1.4 is the corresponding geometric realization of arbitrary finitely presented groups as Out(G) quasi-isometric to the group. This demonstrably breaks the polynomial-or-exponential dichotomy from the known classes, which is the advertised point.

Credit where due: the paper is explicit that it only controls log growth, not the actual norm; that limits precision, but the statement is what it is. The dependence on Ol'shanskii's distortion theorem and Bumagin-Wise is clean, and there is no circularity — the target length functions enter only through Ol'shanskii's construction.

Soft spots: one typo in Definition 1.2(iv) — the bound should be λ^r, not λr, as the paper's own example n↦|n|^α shows. That is cosmetic; nothing in the proofs relies on that clause. The citation to the 'in preparation' paper [8] for the dichotomy in the relative hyperbolic case is a bit unsatisfying as background, but it is explicitly background, not load-bearing. If I had to press for something substantive, it would be that the 'for every nontrivial conjugacy class' in Theorem 1.3 gives equivalence to a linear combination, but the coefficients are not controlled; that is inherent to working up to logarithmic equivalence and not a flaw.

Bottom line: it is a solid short paper, well worth a serious referee. I would send it out. The main theorem is novel at the level of realizability, and the proof is careful. The typo should be caught in revision.

Referee Report

0 major / 5 minor

Summary. The paper studies the growth of conjugacy classes under outer automorphisms of finitely generated groups, measured through the word metric after applying iterates of an automorphism. Its main technical result, Proposition 2.1, shows that in a short exact sequence 1 → N → G → Q → 1 with G non-elementary hyperbolic and N infinite, the natural homomorphism χ: Q → Out(N) is a quasi-isometric embedding when Out(N) is equipped with the Lipschitz metric. The proof combines a simple upper bound (Proposition 2.2) with a lower bound derived from exponential divergence of geodesics in hyperbolic space (Proposition 2.3). Using the Rips construction together with Ol'shanskii's theorem on realising computable length functions as subgroup distortion, the paper proves Theorem 1.3: for any finite list L of computable length functions on Z, there is a finitely generated group G and an outer automorphism Φ such that the log-growth of every nontrivial conjugacy class under Φ is equivalent to a linear combination of the functions in L, and each L ∈ L is realised by some class. Using the Bumagin-Wise strengthening of the Rips construction, the paper also proves Theorem 1.4: every finitely presented group is quasi-isometric, via the Lipschitz metric, to Out(G) for some finitely generated group G.

Significance. If the results hold, they constitute a clean and convincing counterpoint to the polynomial-or-exponential dichotomy that holds for free, abelian, surface, and torsion-free (relatively) hyperbolic groups. The paper shows that once one passes to arbitrary finitely generated groups and works at logarithmic scale, all computable growth types are realisable. The main construction is modular and honest about its external ingredients: the use of Rips' construction, Ol'shanskii's distortion theorem, and the Bumagin-Wise isomorphism variant is explicit and matches the statements of those theorems. The proof of Proposition 2.3 is a careful self-contained hyperbolic-geometry argument, and the upper bound in Proposition 2.2 is a straightforward Lipschitz estimate. The paper also provides a strikingly explicit illustrative example with the Heisenberg group and square-root distortion (Remark 3.5), and it correctly notes that the construction inherits additional properties such as linearity or Kazhdan's property (T) from known variants of the Rips construction.

minor comments (5)

[Definition 1.2(iv)] The bound in condition (iv) should read 'at most λ^r elements' rather than 'at most λr elements'. As written, the displayed example n ↦ |n|^α for α ∈ (0,1) is not a length function, since the ball of radius r has about r^{1/α} elements, which is not bounded linearly in r.
[Proof of Proposition 2.3] Lemma 2.4 is stated for r > 0, but the application sets r = |q|_Q - a - 3δ, which may be nonpositive for small q. The proof should either restrict the argument to q with |q|_Q large and adjust ℓ by a constant, or note that the desired inequality is trivially satisfied for the remaining bounded set of q. This is a local presentation issue and does not affect the validity of the result.
[Theorem 1.3 and Theorem 3.3] The phrase 'an automorphism Φ ∈ Out(G)' should be 'an outer automorphism Φ ∈ Out(G)' for consistency with the standard terminology, since elements of Out(G) are outer automorphism classes and are not literally automorphisms of G.
[References] Reference [8] is cited as 'in preparation' and is used for the background dichotomy in Theorem 1.1. Since Theorem 1.1 is not needed for the main constructions, this is not a load-bearing issue, but the reference should be updated to a published or freely available version if one exists at the time of submission.
[Proof of Theorem 1.3] The phrase 'linear combination of the elements of L' is used where the proof actually establishes equivalence to a sum over the subset of indices for which the corresponding coordinate is nontrivial. The statement could be made slightly more precise by saying 'sum of a subset of the elements of L'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: main theorems are derived from independent external results; the only self-citation is a background citation for Theorem 1.1.

full rationale

The derivation chain for the main results is not circular. Theorem 3.3 takes the target computable length function L as an input and realizes it via Ol'shanskii's Theorem 3.2, an external result that produces a finitely presented group Q and an element q with word length strongly equivalent to L. The Rips construction (with the torsion-free variant) then yields the short exact sequence, and Propositions 2.2 and 2.3 give matching upper and lower bounds: the upper bound comes from the Lipschitz metric and the lower bound from the exponential divergence of the hyperbolic group, so the logarithm of ||Phi^n(c)|| is equivalent to |q^n|_Q. The product construction in Theorem 1.3 is explicit and the 'converse' direction is obtained by taking a class supported on one factor. No equation in the proof assumes the conclusion, and no fitted parameter is relabeled as a prediction. The only self-citation is reference [8], used for the background dichotomy in Theorem 1.1; it is not used in the proofs of Theorems 1.3, 1.4, or 3.3, so it is minor and non-load-bearing. The main load-bearing external ingredients (Rips, Ol'shanskii, Bumagin-Wise) are independent published theorems. Hence the paper earns a low score solely for the presence of that one background self-citation.

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

The paper does not fit parameters to data. It relies on four known construction theorems (Rips, Ol'shanskii, Bumagin-Wise, hyperbolic divergence), all external and cited; no circular dependence on the paper's own conclusions. The free-parameter ledger is empty because all constants in the proof are existential and are shown to exist by the cited theorems.

assumptions (6)

standard math Rips construction: for every finitely presented Q there is a short exact sequence 1→N→G→Q→1 with G hyperbolic and N finitely generated non-elementary; a torsion-free version exists.
Invoked in Theorem 3.1 and in the proof of Theorem 3.3; supplies the kernel N whose automorphism growth is controlled.
standard math Ol'shanskii's distortion theorem: every computable length function on a finitely generated group H is strongly equivalent to the word metric of H inside some finitely presented group.
Theorem 3.2; converts arbitrary computable L on Z into a distorted cyclic subgroup, the input to Theorem 1.3. This is the load-bearing external input for full generality.
standard math Bumagin-Wise strengthening of Rips: one can arrange Q→Out(N) to be an isomorphism.
Used in the proof of Theorem 1.4 to turn the quasi-isometric embedding of Proposition 2.1 into a quasi-isometry.
standard math Exponential divergence of geodesics in hyperbolic spaces, used as Lemma 2.4 from Bridson-Haefliger.
Provides the lower bound ln||Φ(c)|| ≥ κ|q|-ℓ in Proposition 2.3.
standard math Every infinite subgroup of a hyperbolic group contains an infinite-order element, cited from [11, Ch. 8, Cor. 36].
Used in Proposition 2.1 to find a conjugacy class c0 enabling the lower bound for the Lipschitz metric.
standard math Word metrics from different finite generating sets are bi-Lipschitz equivalent, and the Lipschitz metric is well-defined up to quasi-isometry.
Used in the Remark after the definition of the Lipschitz metric; harmless and standard.

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Cite this review

Pith. "Pith review of Examples of groups whose automorphisms have exotic growth." pith.science (2026). https://pith.science/paper/A56BCCFQ

@misc{pith2026190811668,
  author       = {Pith},
  title        = {Pith review of: Examples of groups whose automorphisms have exotic growth},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A56BCCFQ}},
  note         = {Machine review of arXiv:1908.11668}
}

read the original abstract

In this note we produce examples of outer automorphisms of finitely generated groups which have exotic behaviors in terms of growth of conjugacy classes.

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Figures reproduced from arXiv: 1908.11668 by the authors.

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Works this paper leans on

24 extracted references · 22 canonical work pages

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Pith tools

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

[1] [8]

Coulon, A

R. Coulon, A. Hilion, C. Horbez, and G. Levitt. Growth of automorphisms of hyperbolic groups. in preparation, 2019

work page 2019

[2] [1]

I. Agol. The virtual Haken conjecture.Documenta Mathematica, 18:1045– 1087, 2013

work page 2013

[3] [2]

Belegradek and D

I. Belegradek and D. V. Osin. Rips construction and Kazhdan property (T). Groups, Geometry, and Dynamics, 2(1):1–12, 2008. 11

work page 2008

[4] [3]

Bestvina and M

M. Bestvina and M. Handel. Train tracks and automorphisms of free groups. Annals of Mathematics. Second Series, 135(1):1–51, 1992

work page 1992

[5] [4]

M. R. Bridson. Fractional isoperimetric inequalities and subgroup distor- tion. J. Amer. Math. Soc., 12(4):1103–1118, 1999

work page 1999

[6] [5]

M. R. Bridson and A. Haeﬂiger.Metric spaces of non-positive curvature, volume 319 ofGrundlehren der Mathematischen Wissenschaften [Funda- mental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999

work page 1999

[7] [6]

Bumagin and D

I. Bumagin and D. T. Wise. Every group is an outer automorphism group of a ﬁnitely generated group.Journal of Pure and Applied Algebra, 200(1- 2):137–147, 2005

work page 2005

[8] [7]

Coornaert, T

M. Coornaert, T. Delzant, and A. Papadopoulos.Géométrie et théorie des groupes, volume 1441 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 1990

work page 1990