Marked length spectrum rigidity is an important phenomenon in the study of negatively curved and non-positively curved manifolds and in other related contexts. In particular, the Marked Length Spectrum Rigidity Conjecture states that for a closed negatively curved manifold the marked length spectrum (though of as a function on the fundamental group, assigning to each closed curve the length of the shortest element in its free homotopy class) uniquely determines the isometry type of the negatively curved Riemannian metric that gave rise to it.

The Culler-Vogtmann Outer space consists of (equivariant F_N-isometry types of) real trees equipped with free discrete minimal isometric action of F_N. Each such tree defines a translation length function (or marked length spectrum) on F_N, and the tree can be uniquely recovered from its translation length function. We say that a subset S of F_N is "spectrally rigid" in F_N, if whenever two trees from the Outer space have length functions that agree on S, then the two trees are equal in the Outer space.

In this talk we discuss known results about examples of spectrally rigid subsets of free groups, including those coming from random walks and from automorphic orbits.