Spectral rigidity of automorphic orbits in free groups

Abstract

It is well-known that a point $T\in {cv}_N$ in the (unprojectivized) Culler–Vogtmann Outer space ${cv}_N$ is uniquely determined by its translation length function $\parallel \cdot {\parallel }_T:F_N\to ℝ$. A subset $S$ of a free group $F_N$ is called spectrally rigid if, whenever $T,T^{\prime }\in {cv}_N$ are such that $\parallel g{\parallel }_T=\parallel g{\parallel }_{T^{\prime }}$ for every $g\in S$ then $T=T^{\prime }$ in ${cv}_N$. By contrast to the similar questions for the Teichmüller space, it is known that for $N\ge 2$ there does not exist a finite spectrally rigid subset of $F_N$.

In this paper we prove that for $N\ge 3$ if $H\le Aut\left(F_N\right)$ is a subgroup that projects to a nontrivial normal subgroup in $Out\left(F_N\right)$ then the $H$–orbit of an arbitrary nontrivial element $g\in F_N$ is spectrally rigid. We also establish a similar statement for $F_2=F\left(a,b\right)$, provided that $g\in F_2$ is not conjugate to a power of $\left[a,b\right]$.