On the norms of group-invariant transition operators on graphs

In this paper we consider reversible random walks on an infinite grapin, invariant under the action of a closed subgroup of automorphisms which acts with a finite number of orbits on the vertex-set. Thel ²-norm (spectral radius) of the simple random walk is equal to one if and only if the group is both amenable and unimodular, and this also holds for arbitrary random walks with bounded invariant measure. In general, the norm is bounded above by the Perron-Frobenius eigenvalue of a finite matrix, and this bound is attained if and only if the group is both amenable and unimodular.