An integral weight realization theorem for subset currents on free groups

TL;DR

This paper establishes a realization theorem for subset currents on free groups, providing a constructive method to realize prescribed weights via finite graphs, and proves the density of rational subset currents in the space of all subset currents.

Contribution

It introduces a new integral weight realization theorem for subset currents on free groups and offers a direct proof of the density of rational subset currents.

Findings

Realization of weights as occurrences in finite graphs.

Density of rational subset currents in the space of all subset currents.

Answering an open problem about the structure of subset currents.

Abstract

We prove that if $N\ge 2$ and $\alpha: F_N\to \pi_1(\Gamma)$ is a marking on $F_N$, then for any integer $r\ge 2$ and any $F_N$-invariant collection of non-negative integral "weights" associated to all subtrees $K$ of $\widetilde \Gamma$ of radius $\le r$ satisfying some natural "switch" conditions, there exists a finite cyclically reduced folded $\Gamma$-graph $\Delta$ realizing these weights as numbers of "occurrences" of $K$ in $\Delta$. As an application, we give a new, more direct and explicit, proof of one of the main results of our paper with Nagnibeda \cite{KN3} stating that for any $N\ge 2$ the set $\gcnr$ of all rational subset currents is dense in the space $\gcn$ of subset currents on $F_N$. We also answer one of the questions (Problem 10.11) posed in \cite{KN3}. Thus we prove that if a nonzero $\mu\in \gcn$ has all weights with respect to some marking being integers, then $\mu$ is the sum of finitely many "counting" currents corresponding to nontrivial finitely generated subgroups of $F_N$.