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 \emph{translation length function} $||.||_T:F_N\to\mathbb R$. A subset $S$ of a free group $F_N$ is called \emph{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 Teichm\"uller 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 $H\le Aut(F_N)$ 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 $[a,b]$. We also include an appended corrigendum which gives a corrected proof of Lemma 5.1 about the existence of a fully irreducible element in an infinite normal subgroup of of $Out(F_N)$. Our original proof of Lemma 5.1 relied on a subgroup classification result of Handel-Mosher, originally stated by Handel-Mosher for arbitrary subgroups $H\le Out(F_N)$. After our paper was published, it turned out that the proof of the Handel-Mosher subgroup classification theorem needs the assumption that $H$ be finitely generated. The corrigendum provides an alternative proof of Lemma~5.1 which uses the corrected, finitely generated, version of the Handel-Mosher theorem and relies on the 0-acylindricity of the action of $Out(F_N)$ on the free factor complex (due to Bestvina-Mann-Reynolds). A proof of 0-acylindricity is included in the corrigendum.