## Algebraic & Geometric Topology

### Spectral rigidity of automorphic orbits in free groups

#### Abstract

It is well-known that a point $T∈ cvN$ in the (unprojectivized) Culler–Vogtmann Outer space $cvN$ is uniquely determined by its translation length function $∥⋅∥T:FN→ℝ$. A subset $S$ of a free group $FN$ is called spectrally rigid if, whenever $T,T′∈ cvN$ are such that $∥g∥T=∥g∥T′$ for every $g∈S$ then $T=T′$ in $cvN$. By contrast to the similar questions for the Teichmüller space, it is known that for $N≥2$ there does not exist a finite spectrally rigid subset of $FN$.

In this paper we prove that for $N≥3$ if $H≤ Aut(FN)$ is a subgroup that projects to a nontrivial normal subgroup in $Out(FN)$ then the $H$–orbit of an arbitrary nontrivial element $g∈FN$ is spectrally rigid. We also establish a similar statement for $F2=F(a,b)$, provided that $g∈F2$ is not conjugate to a power of $[a,b]$.

#### Article information

Source
Algebr. Geom. Topol., Volume 12, Number 3 (2012), 1457-1486.

Dates
Revised: 19 April 2012
Accepted: 2 May 2012
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715405

Digital Object Identifier
doi:10.2140/agt.2012.12.1457

Mathematical Reviews number (MathSciNet)
MR2966693

Zentralblatt MATH identifier
1261.20040

#### Citation

Carette, Mathieu; Francaviglia, Stefano; Kapovich, Ilya; Martino, Armando. Spectral rigidity of automorphic orbits in free groups. Algebr. Geom. Topol. 12 (2012), no. 3, 1457--1486. doi:10.2140/agt.2012.12.1457. https://projecteuclid.org/euclid.agt/1513715405

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