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We discuss an inequality for graphs, which relates the distances between components of any minimal cut set to the lengths of generators for the homology of the graph. Our motivation arises from percolation theory. In particular this result is applied to Cayley graphs of finite presentations of groups with one end, where it gives an exponential bound on the number of minimal cut sets, and thereby shows that the critical probability for percolation on these graphs is neither zero nor one. We further show for this same class of graphs that the critical probability for the coalescence of all infinite components into a single one is neither zero nor one. Introduction Let G = (V,E) be a locally finite graph. Given two vertices u, v ∈ V , a (u, v) cut set Π ⊆ E is a set of edges that has nonempty intersection with every path from u to v. Similarly a (u,∞) cut set intersects every path from u contained in no finite subgraph. A (u, v) cut set is minimal (an mcs) if it contains no proper subset which is also a (u, v) cut set. Cut sets are naturally connected to homology. We will show how the distance between the two parts of any bipartition of a minimal cut set is related to the L∞ norm on the lattice of integral first cohomology classes and hence also to the L1 norm on homology. In particular we show that no bipartition of a minimal cut set can have its two parts separated by a distance of more than half the diameter of the first homology lattice. For Cayley graphs, this diameter is in turn related to the lengths of the relators. We present the inequality in the context of locally compact, complete, path metric spaces, associating to a graph the path metric in which the length of every edge is one. Our motivation comes from percolation theory [1], [3]. To date percolation has mainly been studied on Z, the Cayley graph for the free Abelian group with d free generators. The basic notion in percolation theory is pc(G), the critical probability for bond percolation on the graph G. In Bernoulli bond percolation the edges are independently colored open with probability p. Those edges that are not open are colored closed. The corresponding product measure on the edge colorings is denoted Pp,G or Pp. For a fixed open/closed edge coloring Eo ∪ Ec = E of a graph G with Received by the editors March 13, 1997. 1991 Mathematics Subject Classification. Primary 60K35. c ©1999 American Mathematical Society

Cut sets and normed cohomology with applications to percolation