Non-amenable products are not treeable

Preprint

5 April 2004

preprint
Published by arXiv in arXiv

https://doi.org/10.48550/arXiv.math/0404096

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.

Keywords

AUTOMORPHISM GROUP
INVARIANT MEASURE
AMENABLE PRODUCTS
INFINITE GRAPHS
TREEABLE
FOREST
ORBIT

All Related Versions

Version 1, 2004-04-05, ArXiv

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