Algebraic & Geometric Topology 12 (2012) 1457–1486

DOI: 10.2140/agt.2012.12.1457

Abstract

It is well-known that a point T in 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′ 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 ≥ 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 in F_N is spectrally rigid. We also establish a similar statement for F₂ = F(a,b), provided that g in F₂ is not conjugate to a power of [a,b].

Keywords

marked length spectrum rigidity, free groups, Outer space

Mathematical Subject Classification

Primary: 20E08, 20F65

Secondary: 53C24, 57M07, 57M50

References

Publication

Received: 3 June 2011
Revised: 19 April 2012
Accepted: 2 May 2012
Published: 10 July 2012

Authors