Subset currents on free groups

We introduce and study the space SCurr(F_N) of subset currents on the free group F_N, and, more generally, on a word-hyperbolic group. A subset current on F_N is a positive F_N-invariant locally finite Borel measure on the space C_N of all closed subsets of ∂F_N consisting of at least two points. The well-studied space Curr(F_N) of geodesics currents-positive F_N-invariant locally finite Borel measures defined on pairs of different boundary points-is contained in the space of subset currents as a closed ℝ-linear Out(F_N)-invariant subspace. Much of the theory of Curr(F_N) naturally extends to the SCurr(F_N) context, but new dynamical, geometric and algebraic features also arise there. While 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 F_N. If a free basis A is fixed in F_N, subset currents may be viewed as F_N-invariant measures on a "branching" analog of the geodesic flow space for F_N, whose elements are infinite subtrees (rather than just geodesic lines) of the Cayley graph of F_N with respect to A. Similarly to the case of geodesics currents, there is a continuous Out(F_N)-invariant "co-volume form" between the Outer space cv_N and the space SCurr (F_N) of subset currents. Given a tree T ∈ cv_N and the "counting current" η_H ∈ SCurr (F_N) corresponding to a finitely generated nontrivial subgroup H ≤ F_N, the value 〈 T, η_H 〉 of this intersection form turns out to be equal to the co-volume of H, that is the volume of the metric graph T_H/H, where T_H ⊆ T is the unique minimal H-invariant subtree of T. However, unlike in the case of geodesic currents, the co-volume form cv_N × SCurr(F_N) → [0,∞) does not extend to a continuous map cv_N × SCurr(F_N) → [0,∞).