`2–torsion of Free-by-cyclic Groups

We provide an upper bound on the `2–torsion of a free-bycyclic group, −ρ(2)(F oΦ Z), in terms of a relative train-track representative for Φ ∈ Aut(F). Our result shares features with a theorem of Lück–Schick computing the `2–torsion of the fundamental group of a 3–manifold that bers over the circle in that it shows that the `2–torsion is determined by the exponential dynamics of the monodromy. In light of the result of Lück–Schick, a special case of our bound is analogous to the bound on the volume of a 3–manifold that bers over the circle with pseudo-Anosov monodromy by the normalized entropy recently demonstrated by Kojima–McShane.