Free-by-Cyclic Groups with Unboundedly Many BNS Component Orbits

An AIM problem asks for families of free-by-cyclic groups with first Betti number greater than two and many connected components of the Bieri–Neumann–Strebel invariant, even after quotienting by the outer automorphism group. For every m ≥ 2 we construct a linearly growing UPG automorphism of F₂ₘ₊₁ whose mapping torus G_m has b₁(G_m) = m+2 and whose BNS invariant is the complement of the m+1 character hyperplanes x+iy = 0 for 0 ≤ i ≤ m. It consequently has exactly 2m+2 connected components. For each component C, we minimize the rank of the kernel of a primitive integral character in C. This minimum is invariant under the full Out(G_m) action. Its m+1 distinct values prove that the components form at least m+1 outer-automorphism orbits. The construction therefore answers the AIM request in the quantitatively unbounded sense. This note is unrefereed and makes no absolute priority claim. Research status: Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-use disclosure: AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text. Scope and priority note: An anonymous public candidate released 2026-08-31 (DOI 10.5281/zenodo.22201487) contains the same construction and conclusions; the manuscript cites it and makes no novelty or public-priority claim.