--- title: "A linearly growing free-by-cyclic family with unbounded BNS component-orbit count" date: 2026-08-31 version: "0.2.0-candidate" doi: 10.5281/zenodo.22201487 pdf: https://github.com/ipitchford/unbounded-bns-component-orbits/releases/download/v0.2.0-candidate/paper.pdf repository: https://github.com/ipitchford/unbounded-bns-component-orbits archive: https://zenodo.org/records/22201487 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # A linearly growing free-by-cyclic family with unbounded BNS component-orbit count ## Summary The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 $$ b_1(G_m)=m+2, \qquad \#\pi_0\!\left(\Sigma^1(G_m)\right)=2m+2, $$ and obtains the orbit lower bound $$ \#\bigl(\pi_0(\Sigma^1(G_m))/\operatorname{Out}(G_m)\bigr)\ge m+1. $$ 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 $$ f_m(e_i)=e_i a_i, \qquad f_m(e_0)=e_0a_0, \qquad f_m(c_i)=c_i a_i^i. $$ Negating the suffix exponents gives an inverse edge-path map, and iterates have linear length. Thus $f_m$ induces a linearly growing automorphism $$ \Phi_m\in\operatorname{Aut}(F_{2m+1}). $$ For its mapping torus $$ G_m=F_{2m+1}\rtimes_{\Phi_m}\mathbb Z, $$ every real character has coordinates $(x,y,z_0,z_2,\ldots,z_m)$ satisfying $$ \chi(t_i)=x+iy, \qquad \chi(a_i)=y\ \ (1\le i\le m), \qquad \chi(a_0)=-my. $$ The stable-letter coordinates are free, so $b_1(G_m)=m+2$. Cashen–Levitt Corollary 2.10 gives $$ \Sigma^1(G_m)= \left\{ [x,y,z_0,z_2,\ldots,z_m]: x+iy\ne0\text{ for }0\le i\le m \right\}. $$ 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, $$ \operatorname{rank}\ker\chi = 1+m|x|+\sum_{i=1}^{m}|x+iy|. $$ 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 $$ \operatorname{rank}\pi_1(\Gamma_m) =(3m+1)-(m+1)+1 =2m+1. $$ The mapping torus becomes a finite graph of groups. Each vertex group is a torus group $$ V_i=\langle a_i,t_i\mid[a_i,t_i]=1\rangle\cong\mathbb Z^2, $$ 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 $$ d^{-1}t_ud=t_va_v^{-p}. $$ The path relations, closing relation and weighted chord relations force the edge-character values to be $$ x,\ x+y,\ \ldots,\ x+my. $$ 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 $$ x+iy=0 \qquad (0\le i\le m). $$ 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: $$ \#\pi_0(\Sigma^1(G_m))=2m+2. $$ 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 $$ \mu(C)= \min\left\{ \operatorname{rank}\ker\chi: \chi:G_m\twoheadrightarrow\mathbb Z,\ [\chi]\in C \right\}. $$ Every chamber is rational and contains primitive integral points, so this minimum exists. If $\alpha\in\operatorname{Aut}(G_m)$, then $$ \ker(\chi\circ\alpha) = \alpha^{-1}(\ker\chi) \cong\ker\chi. $$ 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: $$ \operatorname{rank}\ker\chi = 1+m|x|+\sum_{i=1}^{m}|x+iy|. $$ For the two outer components, every edge value has the same sign. The minimum is $$ \mu(C_{\mathrm{out}}^\pm)=1+2m. $$ For an interior component indexed by $0\le j=2, the BNS description conditional on Cashen-Levitt Corollary 2.10, and the lower bound of m+1 component orbits; not an exact orbit count, a computation of Out(G_m), a classification of free-by-cyclic BNS invariants, a historical-priority finding or external validation. - Reusable methods: Certificate-first, proof-carrying research (certificate-first); Structural compression (structural-compression); Adversarial scientific controls (adversarial-controls); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json - Targeted clocks: assurance, publication, translation - Semantic bridge: explicit — The package records the archived AIM request, the exact Cashen-Levitt dependency, the weighted-chord graph map and graph-of-groups presentation, the character coordinates, the edge forms x+i y, the direct Bass-Serre kernel-rank formula, the integer chamber minima and replay predicates. This exposes every transition from the problem wording to the orbit lower bound. Remaining risks: A producer-side mistake in the graph-of-groups conventions or in checking the hypotheses of Cashen-Levitt Corollary 2.10 could propagate through the proof and replay.; The Python and JavaScript checks share the same producer-defined certificate and do not independently validate the universal argument.; The component invariant separates m+1 classes but does not determine whether equal-minimum components lie in the same Out(G_m)-orbit.; The bounded novelty search cannot establish historical priority.. - Human judgement gates: Keep the exact 2m+2 component count separate from the at-least-m+1 orbit count.; Do not infer that each antipodal component pair is one Out(G_m)-orbit.; Inspect the hypotheses and application of Cashen-Levitt Corollary 2.10 rather than treating the edge-form computation as self-sufficient.; Treat replay for 2<=m<=8 as regression evidence for the formulas, not as proof of the universal theorem.; Keep same-producer cross-language corroboration, internal editorial review, unaffiliated reconstruction, formal verification and peer review distinct.; Make no first-discovery or historical-priority claim from a bounded search. - Next assurance action: Obtain an authenticated unaffiliated reconstruction of the Cashen-Levitt application and Bass-Serre minimum argument, then compute or further constrain Out(G_m) without promoting the present lower bound to an exact orbit count. - Claim ceiling: An anonymous, AI-assisted, unrefereed theorem candidate constructing a linearly growing free-by-cyclic family with exactly 2m+2 BNS components and at least m+1 Out(G_m)-orbits, supported by a complete written proof, one cited structural theorem, public immutable assets, producer replay and internal review; not an exact orbit classification, historical-priority finding, unaffiliated validation, formal verification, authenticated external specialist review or editorial peer review. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — Faster or more reliable construction and assurance of geometric-group-theory examples addressing open problem-list requests in AI-assisted explicit construction, symbolic proof, source correspondence, adversarial replay, internal review and guarded candidate publication; design none; comparator No matched conventional research or publication workflow was registered.; estimand No effect on discovery time, active human effort, compute, correction rate, proof quality, independent-assurance time, uptake, citation or field outcomes was estimated.; no real-world effect evidence asserted - Parent handoffs: depends-on-claim https://web.archive.org/web/20240828224509id_/http://aimpl.org/freebycyclic/6/; inherited claim: AIM Problem 6.2 asks for free-by-cyclic families with b1 greater than two and many BNS components up to the action of Out(G).; inherited ceiling: The archived page fixes the existence request and wording but does not establish this construction, its correctness, novelty, priority or exact orbit count.; extends-result https://doi.org/10.1515/jgth-2015-0038; inherited claim: Cashen and Levitt provide the graph-of-groups BNS criterion used here, a linearly growing circle-of-tori family with many excluded subspheres, and a general fiber-rank formula.; inherited ceiling: Their work supplies the structural criterion and closest located architecture, but not the present weighted-chord family, displayed chamber-minimum sequence or claimed m+1 orbit lower bound. ## 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. ## References 1. American Institute of Mathematics. Rigidity properties of free-by-cyclic groups, BNS invariants, Problem 6.2. Archived 28 August 2024. 2. Cashen, C. H., & Levitt, G. (2016). Mapping tori of free group automorphisms, and the Bieri-Neumann-Strebel invariant of graphs of groups. Journal of Group Theory 19, 191-216. 3. Bieri, R., Neumann, W. D., & Strebel, R. (1987). A geometric invariant of discrete groups. Inventiones Mathematicae 90, 451-477. 4. Funke, F., & Kielak, D. (2018). Alexander and Thurston norms, and the Bieri-Neumann-Strebel invariants for free-by-cyclic groups. Geometry & Topology 22, 2647-2696. 5. Kielak, D. (2020). The Bieri-Neumann-Strebel invariants via Newton polytopes. Inventiones Mathematicae 219, 1009-1068. 6. Andrew, N., & Martino, A. (2022). Free-by-cyclic groups, automorphisms and actions on nearly canonical trees. Journal of Algebra 604, 451-495. 7. Jaikin-Zapirain, A., Kudlinska, M., & Sánchez-Peralta, P. (2026). Thurston norm, polytopes and splitting complexity. arXiv:2606.31774v1. 8. Mutanguha, J. P. (2024). On polynomial free-by-cyclic groups. arXiv:2412.16150v1.