Spectral rigidity of automorphic orbits in free groups

Abstract

It is well-known that a point T 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' in cv_N are such that ||g||_T=||g||_T' for every g in S then T=T' in cv_N. By contrast to the similar questions for the Teichmuller space, it is known that for N > 1 there does not exist a finite spectrally rigid subset of F_N. In this paper we prove that for N > 2 if H < Aut(F_N) is a subgroup that projects to an infinite normal subgroup in Out(F_N) then the H-orbit of an arbitrary nontrivial element g in F_N is spectrally rigid. We also establish a similar statement for F_2=F(a,b), provided that g in F_2 is not conjugate to a power of [a,b].