Subset currents on free groups

Abstract

We introduce and study the space ${{\mathcal{S}{\rm Curr} (F_N)}}$ 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 ${{\mathfrak{C}_N}}$ 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 ${{\mathbb{R}}}$-linear Out(F _N )-invariant subspace. Much of the theory of Curr(F _N ) naturally extends to the ${{\mathcal{S}\;{\rm Curr} (F_N)}}$ 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 ${{\mathcal{S}\;{\rm Curr} (F_N)}}$ of subset currents. Given a tree ${{T \in {\rm cv}_N}}$ and the “counting current” ${{\eta_H \in \mathcal{S}\;{\rm Curr} (F_N)}}$ corresponding to a finitely generated nontrivial subgroup H ≤  F _N , the value ${{\langle T, \eta_H \rangle}}$ 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 ${{T_H \subseteq T}}$ is the unique minimal H-invariant subtree of T. However, unlike in the case of geodesic currents, the co-volume form ${{{\rm cv}_N \times \mathcal{S}\;{\rm Curr}(F_N)\; \to [0,\infty)}}$ does not extend to a continuous map ${{\overline{{\rm cv}}_N \times \mathcal{S}\; {\rm Curr} (F_N) \to [0,\infty)}}$.