Press release · 31 August 2026 · version 0.2.0-candidate
A linearly growing free-by-cyclic family with unbounded BNS component-orbit count
For every m>=2, an explicit linearly growing free-by-cyclic group has first Betti number m+2, exactly 2m+2 BNS components, and at least m+1 component orbits under its outer automorphism group.
Summary
The Bieri–Neumann–Strebel invariant, written $\Sigma^1(G)$, organizes directions in which a finitely generated group maps to the real numbers. Its connected components can distinguish different ways in which a group behaves like a bundle over a circle.
AIM Problem 6.2 asks for free-by-cyclic groups with first Betti number greater than two and many BNS components even after the action of the outer automorphism group is taken into account.
This anonymous, unrefereed candidate gives an explicit family $G_m$, one group for every integer $m\ge 2$. It proves
and obtains the orbit lower bound
The last inequality grows without bound. It is the point needed for the AIM existence request.
The words “at least” are load-bearing. The candidate does not compute $\operatorname{Out}(G_m)$ and does not determine the exact number of component orbits.
Summary for specialists
Let $\Gamma_m$ have vertices $v_0,\ldots,v_m$, a loop $a_i$ at every vertex, path edges $e_i:v_{i-1}\to v_i$, a closing edge $e_0:v_m\to v_0$, and chords $c_i:v_0\to v_i$ for $2\le i\le m$.
The graph map fixes the vertices and loops and sends
Negating the suffix exponents gives an inverse edge-path map, and iterates have linear length. Thus $f_m$ induces a linearly growing automorphism
For its mapping torus
every real character has coordinates $(x,y,z_0,z_2,\ldots,z_m)$ satisfying
The stable-letter coordinates are free, so $b_1(G_m)=m+2$.
Cashen–Levitt Corollary 2.10 gives
The $m+1$ central lines $x+iy=0$ divide the $(x,y)$-plane into exactly $2m+2$ sectors. The extra stable-letter directions do not merge them.
For every primitive integral BNS character,
Minimizing this rank inside each component produces $m+1$ distinct values. Since automorphisms preserve kernel rank, components with different minima cannot lie in the same outer-automorphism orbit.
Technical account: the weighted-chord graph map
The graph $\Gamma_m$ has $m+1$ vertices and $3m+1$ geometric edges. Therefore
The mapping torus becomes a finite graph of groups. Each vertex group is a torus group
and each connecting edge contributes a proper cyclic edge group.
If an oriented connecting edge $d:u\to v$ satisfies $f_m(d)=d a_v^p$, the mapping-torus square gives
The path relations, closing relation and weighted chord relations force the edge-character values to be
The form $x$ occurs on $m$ edge groups: once on the first path edge and once on each of the $m-1$ chords. Each form $x+iy$ for $1\le i\le m$ occurs once.
The chord exponent $i$ is not decorative. It produces the arithmetic progression of edge forms and, later, the separated sequence of chamber minima. One package control replaces these weights by unit weights and requires the claimed certificate to fail.
How the BNS chambers arise
Cashen and Levitt prove that, for the relevant finite reduced graph of groups, a nonzero character lies in $\Sigma^1$ exactly when it is nonzero on every edge group. The hypotheses hold here because every vertex group is $\mathbb Z^2$, the cyclic edge groups are proper, and the splitting is not an ascending HNN extension.
Consequently the deleted set consists of
In the $(x,y)$-plane these are $m+1$ distinct lines through the origin, so their complement has $2m+2$ open convex sectors. A character also has $m$ free stable-letter coordinates, but taking the product with $\mathbb R^m$ does not join different sectors. Positive projectivization preserves the distinction between antipodal directions.
Thus the component count is exact:
This exact component count depends on the cited Cashen–Levitt criterion plus the family-specific edge-form calculation. The release does not claim a new proof of the imported criterion.
The orbit obstruction
A large component count alone does not answer the AIM qualifier “up to $\operatorname{Out}(G)$.” A group automorphism might permute many components into a small number of orbits.
For a component $C$, define
Every chamber is rational and contains primitive integral points, so this minimum exists. If $\alpha\in\operatorname{Aut}(G_m)$, then
Therefore $\mu$ is constant on every $\operatorname{Out}(G_m)$-orbit.
The kernel-rank formula follows directly from the action of $\ker\chi$ on the Bass–Serre tree:
For the two outer components, every edge value has the same sign. The minimum is
For an interior component indexed by $0\le j<m$, write $x=-p$ and $y=q$ with
Exact integer minimization gives the primitive point $(p,q)=(2j+1,2)$ and
These $m$ interior values strictly increase with $j$, and the smallest interior value exceeds the outer value by $m(m-1)$. There are therefore $m+1$ distinct values of $\mu$, which proves at least $m+1$ component orbits.
Small instances make the pattern visible:
| $m$ | $b_1(G_m)$ | BNS components | Proven orbit lower bound | Component-pair minima |
|---|---|---|---|---|
| 2 | 4 | 6 | 3 | $5;\ 7,9$ |
| 3 | 5 | 8 | 4 | $7;\ 13,15,21$ |
| 4 | 6 | 10 | 5 | $9;\ 21,23,29,39$ |
The first number in the last column is the outer-pair minimum; the remaining numbers are the interior-pair minima.
What the lower bound does not show
The argument separates component pairs carrying different minimum kernel ranks. It does not show that two components with the same minimum belong to the same orbit.
In particular, it does not establish that an antipodal pair is one orbit. Depending on the actual automorphism group, the $2m+2$ components could split more finely than the $m+1$ classes detected by $\mu$.
The release therefore does not:
- compute $\operatorname{Out}(G_m)$;
- give an exact component-orbit count;
- classify BNS invariants of all linearly growing free-by-cyclic groups;
- prove a new version of the Cashen–Levitt criterion;
- turn finite replay for $2\le m\le8$ into a proof for all $m$;
- establish historical novelty or priority.
The theorem candidate answers the natural existence request by an unbounded lower bound. It is not an orbit classification.
Evidence, assurance and limitations
The immutable package contains the DOI-bearing bilingual paper, aligned Markdown, a compact proof certificate, exact Python replay, a same-producer language-separated JavaScript reconstruction, eight certificate and claim-boundary controls, source and convention correspondence, citation and bounded novelty audits, the internal editorial record, licences, an environment declaration, checksums and a complete manifest.
The Python verifier recomputes exact graph counts, the rank of the abelianized relation matrix, symbolic character coordinates, edge forms and chamber minima for $2\le m\le8$. It is run normally and with Python optimization. The JavaScript program separately reconstructs the finite edge-form and minimum data using BigInt.
The eight deliberately damaged cases change the free rank, remove a splitting edge, collapse a BNS hyperplane, halve the component count, corrupt an interior minimum, inflate the lower bound into an exact orbit count, replace weighted chords by unit chords, or remove the closing edge from the growth witness. All must be rejected.
One supplied full review was actioned, and one bounded focused confirmation passed on an exact frozen archive. The inherited role-separated reports and the confirmation probe different questions, but they were produced within one coordinated workflow. Reviewer identity, specialist credentials and unaffiliated status were not authenticated.
Public Linux CI reruns the complete read-only package path. GitHub and Zenodo expose the same PDF, 64-member ZIP and manifest; fresh downloads match local SHA-256 values. These facts establish availability, integrity and producer-side replay. They do not establish independent reproduction, independent reimplementation, formal verification, external specialist review, editorial peer review, novelty, priority or field acceptance.
The decisive structural dependency is Cashen–Levitt Corollary 2.10. Correctness still depends on the written verification that its hypotheses apply to this graph of groups.
The Traditional Chinese abstract's formulas were checked for source parity; its linguistic quality has not been independently assessed by a fluent reviewer.
Relationship to earlier work
The general hyperplane-arrangement architecture is not new. Cashen and Levitt prove the graph-of-groups BNS criterion used here. Their Example 5.13 already gives a linearly growing circle-of-tori family requiring many excluded subspheres, and their Theorem 6.1 supplies a general fiber-rank formula.
The narrower candidate contribution is the weighted-chord family, the family-specific direct Bass–Serre rank derivation, and the use of the componentwise minimum
to obstruct the full outer-automorphism action. The candidate rederives the rank formula for this family rather than claiming the general formula as new.
Work by Funke and Kielak, and by Kielak, places BNS invariants and fiber norms in a broader polytope framework. Andrew and Martino study automorphism groups of linearly growing free-by-cyclic groups. Jaikin-Zapirain, Kudlinska and Sánchez-Peralta provide adjacent fixed-rank epimorphism-orbit questions; Mutanguha provides recent polynomial-growth context.
The bounded search found no exact collision for the weighted-chord family or the displayed chamber-minimum sequence. This is not a systematic literature review and does not establish priority.
Who should care, and why
| Audience | Potential use | Required caution |
|---|---|---|
| Geometric group theorists | An explicit test family linking a BNS arrangement to the full outer action | The orbit result is a lower bound, not a classification |
| Researchers studying free-by-cyclic groups | A concrete linearly growing mapping-torus construction with increasing $b_1$ | Cashen–Levitt supplies the decisive BNS criterion |
| Bass–Serre theorists | A direct kernel-rank calculation in which vertex counts cancel and edge multiplicities control rank | The calculation must be checked with the stated orientation conventions |
| Computational reviewers | A compact parametric certificate with exact finite fixtures and hostile mutations | Reimplement independently instead of importing producer predicates |
| Formalizers | A universal graph construction, one external theorem interface and an elementary integer minimization | The imported BNS theorem and its hypotheses need an explicit formal boundary |
| Interested readers | A case where counting regions is not enough because symmetries may identify them | Candidate publication is not field consensus |
Why the problem matters
BNS invariants connect group characters, finiteness properties, splittings and fibrations. Their connected components encode genuinely different regions of character space, but the outer automorphism group can make a large raw count misleading.
The componentwise kernel-rank minimum is useful because it converts the symmetry question into an intrinsic arithmetic obstruction. It does not require a complete description of $\operatorname{Out}(G_m)$ to prove that many components remain inequivalent.
That strategy may be reusable beyond this family: construct a tractable chamber arrangement, attach an automorphism-invariant complexity to each chamber, and force that complexity to take many different values.
How to inspect or reproduce the checks
Use immutable tag v0.2.0-candidate or version DOI 10.5281/zenodo.22201487, not moving main.
For the exact mathematical checks:
python3 -B verify_theorem.py
python3 -B -O verify_theorem.py
python3 -B tests/test_mutations.py
node verify_theorem.mjsFor the complete package, document and manifest gates:
bash run_all.shExpected terminal markers include:
PASS_BNS_FAMILY_CERTIFICATE
PASS_CERTIFICATE_AND_CLAIM_BOUNDARY_NEGATIVE_CONTROLS
PASS_BNS_FAMILY_JAVASCRIPT_RECONSTRUCTION
PASS_READ_ONLY_FROZEN_REPLAYA successful run confirms the encoded finite consequences and package integrity. It does not prove the universal theorem, validate the Cashen–Levitt source independently, establish novelty or confer peer review.
The most valuable next projects
- Compute $\operatorname{Out}(G_m)$ or otherwise determine the exact component-orbit count.
- Decide whether the two members of each antipodal component pair lie in one orbit or two.
- Reconstruct Cashen–Levitt applicability and the Bass–Serre rank argument in a materially separate stack.
- Formalize the graph map, character calculation, imported-theorem interface and chamber minimization.
- Search for broader families in which intrinsic fiber-complexity minima separate still more chamber orbits.
- Obtain authenticated specialist review and a wider historical-priority assessment.
What is in the evidence package
The all-files ZIP contains the DOI-bearing PDF and source, aligned Markdown, the compact proof and machine-readable theorem certificate, exact Python and JavaScript verifiers, eight hostile controls, source/citation/novelty records, review materials and response, licences, environment declaration, release notes, checksums and complete manifest.
The local ZIP is 361,405 bytes with SHA-256 fd1d5d2fb4b0cb25671aa19ea35ec3cf9d38ff879fe0412f67d6d87ce4197640. The version DOI is the citation target. Any mathematical correction should be released as a versioned successor rather than silently replacing the frozen candidate.
Media
The audio briefing is provided in the header above. Download the MP3 briefing · read the transcript.
Open directions for follow-up research
Also available in machine-readable form for research agents and follow-up projects.
- Compute Out(G_m) or otherwise determine the exact number of its orbits on the 2m+2 BNS components; the present theorem proves only a lower bound of m+1.
- Determine whether the two components in each antipodal pair lie in one orbit, two orbits, or a pattern depending on m.
- Independently reconstruct the graph-of-groups presentation, verify the hypotheses of Cashen-Levitt Corollary 2.10 and rederive the kernel-rank minima in a materially separate stack.
- Formalize the graph map, BNS dependency interface, Bass-Serre calculation and integer chamber minimization in a proof assistant.
- Classify broader linearly or polynomially growing free-by-cyclic families with unbounded BNS component-orbit count and obtain authenticated specialist and historical-priority assessments.
Verification status
Anonymous, AI-assisted, unrefereed geometric-group-theory theorem candidate at PASS_WITH_NOTES after actioning a supplied full review and completing one bounded producer-coordinated confirmation. Reviewer identity, specialist credentials and unaffiliated status were not authenticated. The BNS identification imports Cashen-Levitt Corollary 2.10, finite replay is corroborative only, and no exact orbit-count, first-discovery or historical-priority claim is made. The Traditional Chinese abstract's formulas were checked, but its linguistic quality was not independently assessed by a fluent reviewer.
Cite
BibTeX
@misc{unboundedbnscomponentorbits2026,
title = {A linearly growing free-by-cyclic family with unbounded BNS component-orbit count},
author = {Anonymous},
year = {2026},
doi = {10.5281/zenodo.22201487},
url = {https://doi.org/10.5281/zenodo.22201487},
version = {0.2.0-candidate},
howpublished = {Zenodo},
note = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/unbounded-bns-component-orbits/}
}Also: cite.bib · paper.json · this page as Markdown