Subset currents on free groups

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 $A$.

updated version; to appear in Geometriae Dedicata

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• Kesten's theorem for Invariant Random Subgroups
• Lines of minima in Outer space
• Ping-pong and Outer space
• A hyperbolic Out(F_n)-complex
• The actions of $Out(F_k)$ on the boundary of Outer space and on the space of currents: minimal sets and equivariant incompatibility
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• Nonfree Actions of Countable Groups and their Characters

Cited by in corpus (4)

• Invariant random subgroups of linear groups
• An intersection functional on the space of subset currents on a free group
• Currents on cusped hyperbolic surfaces and denseness property
• Subset currents on surfaces