Non-amenable products are not treeable

Details Non-amenable products are not treeable Non-amenable products are not treeable

2004-04-05

arxiv.org/abs/math/0404096v1

Israel J. Math, 118, 147-155 (2000)

Mathematics

Probability

8 pages

Scientific paper

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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