Invariant spanning double rays in amenable groups.

A well-known result of Benjamini, Lyons, Peres, and Schramm states that if G is a finitely generated Cayley graph of a group Γ , then Γ is amenable if and only if G admits a Γ -invariant random spanning tree with at most two ends. We show that this is equivalent to the existence of a Γ -invariant random spanning double ray in a power of G. [ABSTRACT FROM AUTHOR]