Non-amenable products are not treeable

Robin Pemantle; Yuval Peres

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Non-amenable products are not treeable

Robin Pemantle·Yuval Peres

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Abstract

Let X and Y be infinite graphs, such that the automorphism group of X is nonamenable, and the automorphism group of Y has an infinite orbit. We prove that there is no automorphism-invariant measure on the set of spanning trees in the direct product X times Y. This implies that the minimal spanning forest corresponding to i.i.d. edge-weights in such a product, has infinitely many connected components almost surely.

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Robin Pemantle, Yuval Peres. 2004-04-05. Non-amenable products are not treeable. https://arxiv.org/abs/math/0404096

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