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 \(m\geq2\) we construct a linearly growing UPG automorphism of \(F_{2m+1}\) whose mapping torus \(G_m\) has \(b_1(G_m)=m+2\) and whose BNS invariant is the complement of the \(m+1\) character hyperplanes \(x+iy=0\) , \(0\leq i\leq m\) . 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.