Spectral rigidity of automorphic orbits in free groups.

It is well-known that a point T ∈cvN in the (unprojectivized) Culler–Vogtmann Outer space cvN is uniquely determined by its translation length function ∥⋅∥T: FN → ℝ. A subset S of a free group FN is called spectrally rigid if, whenever T,T′∈cvN are such that ∥g∥T = ∥g∥T′ for every g ∈ S then T = T′ in cvN. 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 FN. In this paper we prove that for N ≥ 3 if H ≤Aut(FN) is a subgroup that projects to a nontrivial normal subgroup in Out(FN) then the H–orbit of an arbitrary nontrivial element g ∈ FN 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].