Uniqueness of percolation on products with Z

We show that there exists a connected graph G with subexponential volume growth such that critical percolation on G× Z has infinitely many infinite clusters. We also give some conditions under which this cannot occur. This paper begins with the observation that if G is any connected graph and p is any number in [0, 1], then the number of infinite clusters in p-percolation on G × Z is deterministic, and is either 0, 1 or ∞. The proof is an easy consequence of the fact that one can take any finite set of vertices and translate it along the Z axis and get a set of variables disjoint from the one you started with. In view of this, Sznitman asked whether the argument of Burton and Keane (1989) applies. Namely, assume G is amenable, does it follow that G× Z has only finitely many infinite clusters? The definition of amenability used here is that the Cheeger constant is 0, namely, for every ǫ > 0 there is some finite set of vertices A such that |∂A| ≤ ǫ|A| where ∂A is the edge boundary of A. As stated the answer is no. A binary tree with an infinite path added at the root serves as a counterexample. We suggest a slight modification of this question. Say that G is strongly amenable if G contains no nonamenable subgraph. Assume G is strongly amenable, can one find an interval [p1, p2] such that percolation on G × Z has infinitely many infinite clusters for every p in this interval? What if we further assume that G has polynomial volume growth? Our main result is to construct an example of a strongly amenable graph of the form G × Z with non uniqueness at pc(G × Z). We do not see yet any example of such a graph in which no percolation occurs at pc(G × Z), but non uniqueness occurs for some p > pc(G× Z). Received by the editors July 31, 2012; accepted December 27, 2012. 2010 Mathematics Subject Classification. 60K35.