It is well-known that a point in the (unprojectivized) Culler–Vogtmann Outer space is uniquely determined by its translation length function . A subset of a free group is called spectrally rigid if, whenever are such that for every then in . By contrast to the similar questions for the Teichmüller space, it is known that for there does not exist a finite spectrally rigid subset of .
In this paper we prove that for if is a subgroup that projects to a nontrivial normal subgroup in then the –orbit of an arbitrary nontrivial element is spectrally rigid. We also establish a similar statement for , provided that is not conjugate to a power of .
"Spectral rigidity of automorphic orbits in free groups." Algebr. Geom. Topol. 12 (3) 1457 - 1486, 2012. https://doi.org/10.2140/agt.2012.12.1457