Is there an infinite (finite degree) transitive amenable hyperbolic graph ?



hyperbolic amenable graph
I think the answer is that every such graph should be quasi-isometric to the infinite line. Look at  [this paper.](http://arxiv.org/PS_cache/math/pdf/9806/9806129v1.pdf) Theorem 9 there states that even a quasi-transivite amenable graph should have trivial Poisson boundary (no non-constant harmonic functions). On the other hand, a hyperbolic transitive graph, if it is not quasi-isometric to the Cayley graph $\mathbb Z$, always has non-trivial Poisson boundary. [Here](http://www.cmat.edu.uy/~lessa/tesis/Kaimanovich%20-%20groups%20with%20hyperbolic.pdf) it is proved for groups, but I think it works for transitive hyperbolic graphs as well. 


 **Edit.**  As R W says, the first reference in my answer is not relevant. It is nice that the statement is still true, and that it can be proved using Poisson boundaries. 


