Subset currents on free groups

Ilya Kapovich, Tatiana Nagnibeda

Research output: Contribution to journalArticle

Abstract

We introduce and study the space SCurr(FN) 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 CN 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 ℝ-linear Out(FN)-invariant subspace. Much of the theory of Curr(FN) naturally extends to the SCurr(FN) 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 SCurr (FN) of subset currents. Given a tree T ∈ cvN and the "counting current" ηH ∈ SCurr (FN) corresponding to a finitely generated nontrivial subgroup H ≤ FN, the value 〈 T, ηH 〉 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 TH ⊆ T is the unique minimal H-invariant subtree of T. However, unlike in the case of geodesic currents, the co-volume form cvN × SCurr(FN) → [0,∞) does not extend to a continuous map cvN × SCurr(FN) → [0,∞).

Original languageEnglish (US)
Pages (from-to)307-348
Number of pages42
JournalGeometriae Dedicata
Volume166
Issue number1
DOIs
StatePublished - 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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