Spectral rigidity of automorphic orbits in free groups

Abstract

It is well-known that a point $T\in c{v}_N$ in the (unprojectivized) Culler-Vogtmann Outer space $c{v}_N$ 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 $T,T^{\prime }\in c{v}_N$ are such that $||g|{|}_T=||g|{|}_{T^{\prime }}$ for every $g\in S$ then $T=T^{\prime }$ in $c{v}_N$. 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 $H\le Aut(F_N)$ 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.

Included a 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). Note that, because of the arXiv rules, the corrigendum and the original article are amalgamated into a single pdf file, with the corrigendum appearing first, followed by the main body of the original article

References in corpus (3)

• $\mathbb{R}$-trees and laminations for free groups II: The dual lamination of an $\mathbb{R}$-tree
• Geometric Intersection Number and analogues of the Curve Complex for free groups
• Intersection form, laminations and currents on free groups

Cited by in corpus (1)

• Generalized north-south dynamics on the space of geodesic currents