  An important conjecture in percolation theory is that almost surely no
infinite cluster exists in critical percolation on any transitive graph for
which the critical probability is less than 1. Earlier work has established
this for the amenable cases Z^2 and Z^d for large d, as well as for all
non-amenable graphs with unimodular automorphism groups. We show that the
conjecture holds for the basic classes of non-amenable graphs with
non-unimodular automorphism groups: for decorated trees and the non-unimodular
Diestel-Leader graphs. We also show that the connection probability between two
vertices decay exponentially in their distance. Finally, we prove that critical
percolation on the positive part of the lamplighter group has no infinite
clusters.
