The Free Uniform Spanning Forest is disconnected in some virtually free groups, depending on the generator set

TL;DR

This paper demonstrates that the structure of the Free Uniform Spanning Forest (FUSF) can vary dramatically in certain virtually free groups, depending on the generating set, challenging previous assumptions about invariance properties.

Contribution

It provides counterexamples showing FUSF properties depend on the generating set and group structure, disproving conjectures and answering open questions.

Findings

FUSF can have infinitely many trees in certain graph products

FUSF properties are not quasi-isometry invariants

Counterexamples to conjectures about unimodular graphs and FUSF

Abstract

We prove the rather counterintuitive result that there exist finite transitive graphs H and integers k such that the Free Uniform Spanning Forest in the direct product of the k-regular tree and H has infinitely many trees almost surely. This shows that the number of trees in the FUSF is not a quasi-isometry invariant. Moreover, we give two different Cayley graphs of the same virtually free group such that the FUSF has infinitely many trees in one, but is connected in the other, answering a question of Lyons and Peres (2016) in the negative. A version of our argument gives an example of a non-unimodular transitive graph where WUSF\not=FUSF, but some of the FUSF trees are light with respect to Haar measure. This disproves a conjecture of Tang (2019).