Spectral rigidity of automorphic orbits in free groups

Carette, Mathieu;Stefano, Francaviglia;Kapovich, Ilya;Martino, Armando
(2012) Algebraic & Geometric Topology — Vol. 12, n° 3, p. 1457-1486 (2012)

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Authors
  • Carette, MathieuUCLouvain
    Author
  • Stefano, FrancavigliaUniversity of Bologna
    Author
  • Kapovich, IlyaUniversity of Illinois at Urbana-Champaign
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  • Martino, ArmandoUniversity of Southampton
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Abstract
It is well-known that a point T 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' in cv_N are such that ||g||_T=||g||_T' for every g in S then T=T' in cv_N. By contrast to the similar questions for the Teichmüller space, it is known that for N > 1 there does not exist a finite spectrally rigid subset of F_N. In this paper we prove that for N > 2 if H < Aut(F_N) is a subgroup that projects to an infinite normal subgroup in Out(F_N) 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(a,b), provided that g in F_2 is not conjugate to a power of [a,b].
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Carette, M., Stefano, F., Kapovich, I., & Martino, A. (2012). Spectral rigidity of automorphic orbits in free groups. Algebraic & Geometric Topology, 12(3), 1457-1486. https://doi.org/10.2140/agt.2012.12.1457 (Original work published 2012)