论文标题
根据发电机组，跨越森林的自由统一森林几乎没有连接
The Free Uniform Spanning Forest is disconnected in some virtually free groups, depending on the generator set
论文作者
论文摘要
我们证明存在相当违反直觉的结果，即存在有限的横向图H和整数k，使得跨越K型树和H的直接产物中的自由统一森林几乎肯定地肯定有许多树木。 这表明FUSF中的树的数量并不是准偶然的不变。此外，我们给出了同一几乎没有免费组的两个不同的Cayley图，以使FUSF中有无限的树，但另一个是相互联系的，在负面回答了Lyons and Peres（2016）的问题。 我们的论点的一个版本给出了一个非较小的及物图的示例，其中wusf \ not = fusf，但与HAAR度量相对于HAAR测量，某些FUSF树是轻的。这反驳了唐（2019）的猜想。
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).
