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

We prove the rather counterintuitive result that there exist finite transitive graphs H and integers k such Free Uniform Spanning Forest in direct product of k-regular tree has infinitely many trees almost surely. This shows number FUSF is not a quasi-isometry invariant. Moreover, we give two different Cayley same virtually free group one, but connected other, answering question Lyons Peres (Probability on Trees Networks (2016) Cambridge Univ. Press) negative. A version our argument gives an example nonunimodular graph where WUSF≠FUSF, some are light with respect to Haar measure. disproves conjecture Tang (Electron. J. Probab. 26 (2021) Paper No. 141).