Geometriae Dedicata

, Volume 166, Issue 1, pp 307–348

Subset currents on free groups

Authors

    • Department of MathematicsUniversity of Illinois at Urbana-Champaign
  • Tatiana Nagnibeda
    • Section de mathématiquesUniversité de Genève
Original Paper

DOI: 10.1007/s10711-012-9797-y

Cite this article as:
Kapovich, I. & Nagnibeda, T. Geom Dedicata (2013) 166: 307. doi:10.1007/s10711-012-9797-y
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Abstract

We introduce and study the space \({{\mathcal{S}{\rm Curr} (F_N)}}\) of subset currents on the free group FN, and, more generally, on a word-hyperbolic group. A subset current on FN is a positive FN-invariant locally finite Borel measure on the space \({{\mathfrak{C}_N}}\) of all closed subsets of ∂FN consisting of at least two points. The well-studied space Curr(FN) of geodesics currents–positive FN-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(FN)-invariant subspace. Much of the theory of Curr(FN) 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 FN. If a free basis A is fixed in FN, subset currents may be viewed as FN-invariant measures 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. Similarly to the case of geodesics currents, there is a continuous Out(FN)-invariant “co-volume form” between the Outer space cvN 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 ≤  FN, 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 TH/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 \({{\overline{{\rm cv}}_N \times \mathcal{S}\; {\rm Curr} (F_N) \to [0,\infty)}}\).

Keywords

Free groupsGeodesic currentsOuter spaceAutomorphisms of free groups

Mathematics Subject Classification (2000)

Primary 20FSecondary 57M37B37D
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© Springer Science+Business Media Dordrecht 2012