### Abstract

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,∞).

Original language | English (US) |
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Pages (from-to) | 307-348 |

Number of pages | 42 |

Journal | Geometriae Dedicata |

Volume | 166 |

Issue number | 1 |

DOIs | |

State | Published - Oct 1 2013 |

### Keywords

- Automorphisms of free groups
- Free groups
- Geodesic currents
- Outer space

### ASJC Scopus subject areas

- Geometry and Topology

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## Cite this

*Geometriae Dedicata*,

*166*(1), 307-348. https://doi.org/10.1007/s10711-012-9797-y