Subset currents on free groups

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We introduce and study the space of subset currents on the free group Fₙ. A subset current on Fₙ is a positive Fₙ-invariant locally finite Borel measure on the space Cₙ of all closed subsets of ∂ Fₙ consisting of at least two points. While ordinary geodesic currents generalize conjugacy classes of nontrivial group elements, a subset current is a measure-theoretic generalization of the conjugacy class of a nontrivial finitely generated subgroup in Fₙ, and, more generally, in a word-hyperbolic group. The concept of a subset current is related to the notion of an "invariant random subgroup" with respect to some conjugacy-invariant probability measure on the space of closed subgroups of a topological group. If we fix a free basis A of Fₙ, a subset current may also be viewed as an Fₙ-invariant measure on a "branching" analog of the geodesic flow space for Fₙ, whose elements are infinite subtrees (rather than just geodesic lines) of the Cayley graph of Fₙ with respect to A.