## 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 (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 WUSF ≠ FUSF, but some of the FUSF trees are light with respect to Haar measure. This disproves a conjecture of Tang (Electron. J. Probab. 26 (2021) Paper No. 141).

Original language | English |
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Pages (from-to) | 2218-2243 |

Number of pages | 26 |

Journal | Annals of Probability |

Volume | 50 |

Issue number | 6 |

DOIs | |

Publication status | Published - Nov 2022 |

### Bibliographical note

Funding Information:The first author is also at the Institute of Mathematics, Budapest University of Technology and Economics. The second author is also at the Alfréd Rényi Institute of Mathematics, Budapest. Our work was supported by the ERC Consolidator Grant 772466 “NOISE.” The second author was partially supported by Icelandic Research Fund Grant 185233-051.

Funding Information:

Our work was supported by the ERC Consolidator Grant 772466 “NOISE.” The second author was partially supported by Icelandic Research Fund Grant 185233-051.

Publisher Copyright:

© Institute of Mathematical Statistics, 2022

## Other keywords

- Free uniform spanning forest
- Nonunimodular transitive graphs
- Nonuniversality at criticality.
- Virtually free groups
- Wilson’s algorithm