We develop methods for studying transition operators on metric spaces that are invariant under a co-compact group which acts properly. A basic requirement is a decomposition of such operators with respect to the group orbits. We then introduce "reduced" transition operators on the compact factor space whose norms and spectral radii are upper bounds for the -norms and spectral radii of the original operator. If the group is amenable then the spectral radii of the original and reduced operators coincide, and under additional hypotheses, this is also sufficient for amenability. Further bounds involve the modular function of the group. In this framework, we prove among other things that the bottom of the spectrum of the Laplacian on a co-compact Riemannian manifold is 0 if and only if the group is amenable and unimodular. The same result holds for Euclidean simplicial complexes. On a geodesic, proper metric space with co-compact isometry group action, the averaging operator over balls with a fixed radius has norm equal to 1 if and only if the group is amenable and unimodular. The technique also allows explicit computation of spectral radii when the group is amenable.
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Laurent Saloff-Coste, Wolfgang Woess, Transition operators on co-compact -spaces. Rev. Mat. Iberoam. 22 (2006), no. 3, pp. 747–799DOI 10.4171/RMI/473