Spectral rigidity of automorphic orbits in free groups

Carette, Mathieu¹ ; Francaviglia, Stefano² ; Kapovich, Ilya³ ; Martino, Armando⁴

¹SST/IRMP, Chemin du Cyclotron 2, bte L7.01.01, 1348 Louvain-la-Neuve, Belgium
²Dipartimento di Matematica of the University of Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy
³Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana IL 61801, USA
⁴School of Mathematics, University of Southampton, Highfield, Southampton, SO17 1BJ, United Kingdom

Algebraic and Geometric Topology, Tome 12 (2012) no. 3, pp. 1457-1486

Résumé

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].

Détail

DOI : 10.2140/agt.2012.12.1457

Classification : 20E08, 20F65, 57M07, 57M50, 53C24
Keywords: marked length spectrum rigidity, free groups, Outer space

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