Subset currents on free groups

TL;DR

This paper introduces the concept of subset currents on free groups, extending geodesic currents to measures on closed subsets of the boundary, capturing subgroup structures in a measure-theoretic framework.

Contribution

It defines and studies subset currents as a new measure-theoretic tool generalizing conjugacy classes and subgroup structures in free and hyperbolic groups.

Findings

Defines subset currents as measures on closed subsets of the boundary

Establishes the relation to invariant random subgroups

Connects subset currents to a branching analog of geodesic flow

Abstract

We introduce and study the space of \emph{subset currents} on the free group $F_N$. 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 $\partial F_N$ consisting of at least two points. While ordinary 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 $F_N$, and, more generally, in a word-hyperbolic group. The concept of a subset current is related to the notion of an "invariant random subgroup" with respect to some conjugacy-invariant probability measure on the space of closed subgroups of a topological group. If we fix a free basis $A$ of $F_N$, a subset current may also be viewed as an $F_N$-invariant measure 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 $A$.