自由均匀生成森林在某些虚拟自由群中是不连通的，这取决于生成元集
The Free Uniform Spanning Forest is disconnected in some virtually free groups, depending on the generator set
摘要：
证明了一个相当违反直觉的结果：存在有限传递图H和整数k，使得k-正则树与H的直积中的自由均匀生成森林几乎必然有无穷多棵树.这表明FUSF中的树的数目不是准等距不变量。此外，我们给出了同一个虚拟自由群的两个不同的Cayley图，使得FUSF在一个中有无限多棵树，但在另一个中是连通的，回答了Lyons和Peres（2016）的一个问题。我们的论证的一个版本给出了一个非幺模传递图的例子，其中WUSF \not =FUSF，但某些FUSF树相对于Haar测度是轻的。这证明了Tang（2019）的一个猜想是错误的。
英文摘要：
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 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 non-unimodular 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).
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