{"id":"2835c5f6-fd76-4ef8-9135-1842b423b8df","arxiv_id":"1908.11668","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Every computable length function on Z is, up to equivalence, the logarithmic growth of a conjugacy class under an outer automorphism of a finitely generated group.","lead":"Finite collections of any computable growth rules on the integers can be realized as how fast conjugacy classes grow under an outer automorphism of some finitely generated group. The paper matters because it shows the simple polynomial-or-exponential pattern seen in free and hyperbolic groups fails completely for general groups.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"After reading the paper and the reader's verdict, I find no load-bearing concern. The proof of Proposition 2.1 is complete: Proposition 2.2 gives the upper bound, and Proposition 2.3 gives the lower bound via exponential divergence and a clever minimization over preimages. The step where a point x near the identity is found on a geodesic is justified by quadrilateral thinness, and the constants depend only on the fixed conjugacy class and the hyperbolic space. Theorem 3.3 combines this with Ol'shanskii's distortion theorem; the torsion-free Rips construction guarantees every non-trivial class in N has infinite order, so Proposition 2.3 applies to all of them. The product construction for Theorem 1.3 is correct because the word length in a direct product with the union generating set is the sum of factor lengths, so the logarithm of the norm of a class with support I is equivalent to the maximum, hence to the sum, of the corresponding L_i. The only external input, Ol'shanskii's Theorem 3.2, is a standard published result; the paper's reliance on it is explicit and appropriate. I therefore concur with the reader's ACCEPT. Minor typographical issue: Definition 1.2(iv) should read λ^r rather than λr, otherwise the example n↦|n|^α for α<1 would violate the definition.","tokens_in":8865,"tokens_out":37024,"duration_ms":320151,"concrete_test":"Independently re-derive Proposition 2.3 for the explicit Rips pair in Remark 3.5, numerically confirming that ln‖Φ^n(c)‖ is comparable to √n for a non-trivial conjugacy class c; this exercises the exponential-divergence lower bound and the torsion-free Rips hypothesis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is well-supported. Proposition 2.1 is proved in detail: the upper bound is a triangle-inequality argument, and the lower bound uses the exponential divergence of the hyperbolic group together with a minimization argument over preimages. The application to Theorem 1.3 passes through Ol'shanskii's Theorem 3.2 and the product construction; the log-norm of a product class is indeed equivalent to the sum of the relevant L_i. The only external premise that could threaten individual instances is Ol'shanskii's Theorem 3.2, but it is a published theorem and the paper's use of it matches its statement. The torsion-free Rips variant ensures every non-trivial element of N has infinite order, which Proposition 2.3 requires. I found no internal inconsistency or omitted step. A small typographical issue exists in Definition 1.2(iv): the bound should be 'at most λ^r elements' rather than 'λr', as shown by the paper's own example n↦|n|^α for α∈(0,1).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":8984,"tokens_out":12422,"duration_ms":120932,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"Definition 1.2(iv)"},{"comment":"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.","section":"Proof of Proposition 2.3"},{"comment":"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.","section":"Theorem 1.3 and Theorem 3.3"},{"comment":"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.","section":"References"},{"comment":"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'.","section":"Proof of Theorem 1.3"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nCredit 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.\n\nSoft 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.\n\nBottom 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.","headline":"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.","tokens_in":9556,"tokens_out":5206,"would_cite":true,"duration_ms":47116,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20E36","20F28","20F65","20F67","20F06"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["growth of automorphisms","outer automorphism groups","conjugacy classes","hyperbolic groups","Rips construction","distortion of subgroups","length functions","Lipschitz metric"],"falsifier":"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.","tokens_in":8603,"feed_emoji":"📈","tokens_out":10131,"duration_ms":76833,"temperature":0.7,"pith_summary":"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.","feed_headline":"Automorphisms can grow in any computable pattern","feed_subtitle":"Every computable length function appears as logarithmic conjugacy-class growth, breaking the old dichotomy.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Establishes the distortion theorem used to realize any computable length function on Z as the word length of an element in a finitely presented group; without it the construction cannot import arbitrary L.","marker":"[18]"},{"why":"Provides the Rips construction, the short exact sequence with G hyperbolic and N finitely generated non-elementary, on which the quasi-isometric embedding result rests.","marker":"[20]"},{"why":"Supplies the exponential-divergence estimate for hyperbolic spaces, reformulated as Lemma 2.4, that gives the lower bound on the Lipschitz metric.","marker":"[5]"},{"why":"Refines the Rips construction so that Q -> Out(N) is an isomorphism, yielding Theorem 1.4 and showing the Lipschitz metric can realize any finitely presented group.","marker":"[6]"},{"why":"Provides the hyperbolic-geometry background used to show that the set A(u) of elements moving u by at most a constant is quasi-isometric to a line, a step in the proof of Proposition 2.3.","marker":"[7]"}],"fun_headline_variants":["Outer automorphisms realize any computable growth","Computable patterns for conjugacy-class growth","Automorphism growth: any computable pattern","Outer automorphisms: computable conjugacy-class growth"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Outer automorphisms realize any computable growth","Computable patterns for conjugacy-class growth","Automorphism growth: any computable pattern","Outer automorphisms: computable conjugacy-class growth"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000484,"raw_usage":{"total_tokens":2298,"prompt_tokens":761,"completion_tokens":1537,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":377,"completion_tokens_details":{"reasoning_tokens":1476}},"tokens_in":377,"tokens_out":1537,"duration_ms":11050,"temperature":1.0,"reasoning_tokens":1476,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:09:26.148728+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the distortion theorem used to realize any computable length function on Z as the word length of an element in a finitely presented group; without it the construction cannot import arbitrary L."},{"cited_title":"Subgroupsofsmallcancellationgroups","cited_arxiv_id":null,"evidence_quote":"Provides the Rips construction, the short exact sequence with G hyperbolic and N finitely generated non-elementary, on which the quasi-isometric embedding result rests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the exponential-divergence estimate for hyperbolic spaces, reformulated as Lemma 2.4, that gives the lower bound on the Lipschitz metric."},{"cited_title":"Bumagin and D","cited_arxiv_id":null,"evidence_quote":"Refines the Rips construction so that Q -> Out(N) is an isomorphism, yielding Theorem 1.4 and showing the Lipschitz metric can realize any finitely presented group."},{"cited_title":"Coornaert, T","cited_arxiv_id":null,"evidence_quote":"Provides the hyperbolic-geometry background used to show that the set A(u) of elements moving u by at most a constant is quasi-isometric to a line, a step in the proof of Proposition 2.3."}],"review_version":1}