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

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 $\mathsf{FUSF}$ is not a quasi-isometry invariant. Moreover, we give two different Cayley graphs of the same virtually free group such that the $\mathsf{FUSF}$ has infinitely many trees in one, but is connected in the other, answering a question of Lyons and Peres (Probability on Trees and Networks (2016) Cambridge Univ. Press) in the negative.

A version of our argument gives an example of a nonunimodular transitive graph where $\mathsf{WUSF}\ne \mathsf{FUSF}$, but some of the $\mathsf{FUSF}$ trees are light with respect to Haar measure. This disproves a conjecture of Tang (Electron. J. Probab. 26 (2021) Paper No. 141).