Non-amenable products are not treeable
Probability
2007-05-23 v1 Geometric Topology
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
Cite
@article{arxiv.math/0404096,
title = {Non-amenable products are not treeable},
author = {Robin Pemantle and Yuval Peres},
journal= {arXiv preprint arXiv:math/0404096},
year = {2007}
}
Comments
8 pages