Spectral rigidity of automorphic orbits in free groups

It is well-known that a point T ∊ cv_N in the (unprojectivized) Culler-Vogtmann Outer space cv_N is uniquely determined by its translation length function || · ||_T: F_N → ℝ. A subset S of a free group F_N is called spectrally rigid if, whenever T, T′ ∊ cv_N are such that ||g||_T = ||g||_T′ for every g ∊ S then T = T′ in cv_N. By contrast to the similar questions for the Teichmüller space, it is known that for N ≥ 2 there does not exist a finite spectrally rigid subset of F_N. In this paper we prove that for N ≥ 3 if H ≤ Aut(F_N) is a subgroup that projects to a nontrivial normal subgroup in Out(F_N) then the H-orbit of an arbitrary nontrivial element g ∊ F_N is spectrally rigid. We also establish a similar statement for F₂ = F(a, b), provided that g ∊ F₂ is not conjugate to a power of [a, b].