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 kregular tree and H has infinitely many trees almost surely. This shows that the number of trees in the FUSF is not a quasiisometry 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 nonunimodular 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).
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 DOI:
 10.48550/arXiv.2006.06387
 arXiv:
 arXiv:2006.06387
 Bibcode:
 2020arXiv200606387P
 Keywords:

 Mathematics  Probability;
 Mathematics  Group Theory
 EPrint:
 27 pages, 4 figures. Important corrections in the proofs (Sections 3 and 4), small changes in Sections 1 and 6