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].
- Free groups
- Marked length spectrum rigidity
- Outer space
ASJC Scopus subject areas
- Geometry and Topology