Component count on nonunimodular transitive networks

Determine whether the free spanning forest on every transitive nonunimodular network has either one component or infinitely many components.

Background

The paper proves that the number of components of the free spanning forest is almost surely constant on every connected, locally finite, infinite network. This does not determine which constant occurs.

A stronger dichotomy—one component or infinitely many components—is known for unimodular transitive graphs, but the paper explicitly records that the corresponding question remains unresolved for transitive networks in the nonunimodular case.

References

We also remark that whether the FSF on every transitive network has either one or infinitely many components Question~15.6, Question~10.29 remains open in the nonunimodular case; the unimodular case was settled by Hutchcroft and Nachmias and independently by Timar .

— A zero-one law for swap-invariant events in uniform spanning forests  (2610.00938 - Li et al., 1 Oct 2026) in Section 3, paragraph titled “Number of components”