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Apr 3, 2011 at 16:37 comment added Andreas Thom Thanks a lot! I did not know about this paper. Kevin Whyte's paper ("Amenability, bi-Lipschitz equivalence, and the von Neumann conjecture." Duke Math. J. 99 (1999), no. 1, 93–112) appeared two years later.
Apr 3, 2011 at 15:52 comment added Al Tal I've made a small search and found the paper by Benjamini and Schramm "Every Graph With A Positive Cheeger Constant Contains A Tree With A Positive Cheeger Constant", GAFA, 1997. The mentioned result is a consequence of their theorem 1.5, see also the Remark 1.9.
Apr 3, 2011 at 15:23 history edited Al Tal CC BY-SA 2.5
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Apr 3, 2011 at 15:18 history edited Al Tal CC BY-SA 2.5
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Apr 3, 2011 at 14:44 comment added Andreas Thom This was proved by Kevin Whyte for non-amenable groups.
Apr 3, 2011 at 14:37 history answered Al Tal CC BY-SA 2.5