### Abstract

It is well-known that a point T ∊ cv_{N} in the (unprojectivized) Culler-Vogtmann Outer space cv_{N} is uniquely determined by its translation length function || · ||_{T}: F_{N} → ℝ. A subset S of a free group F_{N} is called spectrally rigid if, whenever T, T′ ∊ cv_{N} are such that ||g||_{T} = ||g||_{T′} for every g ∊ S then T = T′ in cv_{N}. 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 F_{N}. In this paper we prove that for N ≥ 3 if H ≤ Aut(F_{N}) is a subgroup that projects to a nontrivial normal subgroup in Out(F_{N}) then the H-orbit of an arbitrary nontrivial element g ∊ F_{N} is spectrally rigid. We also establish a similar statement for F_{2} = F(a, b), provided that g ∊ F_{2} is not conjugate to a power of [a, b].

Original language | English (US) |
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Pages (from-to) | 1457-1486 |

Number of pages | 30 |

Journal | Algebraic and Geometric Topology |

Volume | 12 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2012 |

### Keywords

- Free groups
- Marked length spectrum rigidity
- Outer space

### ASJC Scopus subject areas

- Geometry and Topology

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## Cite this

*Algebraic and Geometric Topology*,

*12*(3), 1457-1486. https://doi.org/10.2140/agt.2012.12.1457