Algebraic & Geometric Topology

Spectral rigidity of automorphic orbits in free groups

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

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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. A subset S of a free group FN is called spectrally rigid if, whenever T,T cvN are such that gT=gT for every gS then T=T in cvN. By contrast to the similar questions for the Teichmüller space, it is known that for N2 there does not exist a finite spectrally rigid subset of FN.

In this paper we prove that for N3 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 gFN is spectrally rigid. We also establish a similar statement for F2=F(a,b), provided that gF2 is not conjugate to a power of [a,b].

Article information

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

Received: 3 June 2011
Revised: 19 April 2012
Accepted: 2 May 2012
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20E08: Groups acting on trees [See also 20F65] 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 57M07: Topological methods in group theory 57M50: Geometric structures on low-dimensional manifolds 53C24: Rigidity results

marked length spectrum rigidity free groups Outer space


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.

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