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Nonamenable products are not treeable. (English) Zbl 0961.43002

This is a nice application of amenability of groups to graph theory. Let \(X\) and \(Y\) be infinite graphs and let \(\text{Aut}(X)\) and \(\text{Aut}(Y)\) be their automorphism groups, respectively. The authors prove that if \(\text{Aut}(X)\) is not amenable and \(\text{Aut}(Y)\) has an infinite orbit, then there is no automorphism-invariant measure on the set of spanning trees in the direct product \(X\times Y\). This result implies that the minimal spanning forest corresponding to i.i.d. edge-weights in \(X\times Y\) has infinitely many connected components almost surely.

MSC:

43A07 Means on groups, semigroups, etc.; amenable groups
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