, Volume 166, Issue 1, pp 307-348
Date: 17 Nov 2012

Subset currents on free groups

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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 \({{\overline{{\rm cv}}_N \times \mathcal{S}\; {\rm Curr} (F_N) \to [0,\infty)}}\) .

The first author was supported by the NSF grant DMS-0904200. Both authors acknowledge the support of the Swiss National Foundation for Scientific Research.