#### Volume 12, issue 3 (2012)

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Spectral rigidity of automorphic orbits in free groups

### Mathieu Carette, Stefano Francaviglia, Ilya Kapovich and Armando Martino

Algebraic & Geometric Topology 12 (2012) 1457–1486
##### 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 $\parallel \cdot {\parallel }_{T}:\phantom{\rule{0.3em}{0ex}}{F}_{N}\to ℝ$. 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 $\left[a,b\right]$.

##### Keywords
marked length spectrum rigidity, free groups, Outer space
##### Mathematical Subject Classification 2010
Primary: 20E08, 20F65
Secondary: 57M07, 57M50, 53C24
##### Publication
Revised: 19 April 2012
Accepted: 2 May 2012
Published: 10 July 2012
##### Authors
 Mathieu Carette SST/IRMP Chemin du Cyclotron 2 bte L7.01.01 1348 Louvain-la-Neuve Belgium Stefano Francaviglia Dipartimento di Matematica of the University of Bologna Piazza di Porta S. Donato 5 40126 Bologna Italy Ilya Kapovich Department of Mathematics University of Illinois at Urbana-Champaign 1409 West Green Street Urbana IL 61801 USA Armando Martino School of Mathematics University of Southampton Highfield Southampton SO17 1BJ United Kingdom