Spectral rigidity of automorphic orbits in free groups

It is well-known that a point T 2 cvN in the (unprojectivized) Culler–Vogtmann Outer space cvN is uniquely determined by its translation length function k kT W FN !R . A subset S of a free group FN is called spectrally rigid if, whenever T;T 0 2 cvN are such that kgkT D kgkT 0 for every g 2 S then T D T 0 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 2 FN is spectrally rigid. We also establish a similar statement for F2 D F.a; b/ , provided that g 2 F2 is not conjugate to a power of Œa; b .