THE FREE UNIFORM SPANNING FOREST IS DISCONNECTED IN SOME VIRTUALLY FREE GROUPS, DEPENDING ON THE GENERATOR SET

• 成果类型: Article
• 署名作者: Pete, Gabor; Timar, Adam
• 署名单位: HUN-REN; HUN-REN Alfred Renyi Institute of Mathematics; University of Iceland
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1581
• 发表日期: 2022
• 页码: 2218-2243
• 关键词: harmonic-functions; interlacements; components; clusters

摘要

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 virtu-ally 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 tran-sitive graph where WUSF not equal 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).