# Transition operators on co-compact G-spaces.

Laurent Saloff-Coste; Wolfgang Woess

Revista Matemática Iberoamericana (2006)

- Volume: 22, Issue: 3, page 747-799
- ISSN: 0213-2230

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topSaloff-Coste, Laurent, and Woess, Wolfgang. "Transition operators on co-compact G-spaces.." Revista Matemática Iberoamericana 22.3 (2006): 747-799. <http://eudml.org/doc/41991>.

@article{Saloff2006,

abstract = {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 Lp-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.},

author = {Saloff-Coste, Laurent, Woess, Wolfgang},

journal = {Revista Matemática Iberoamericana},

keywords = {Cadenas de Markov; Probabilidad de transición; Operador laplaciano; G-espacios; Invariancia; -space; proper action; isometries; invariant transition operator; amenability},

language = {eng},

number = {3},

pages = {747-799},

title = {Transition operators on co-compact G-spaces.},

url = {http://eudml.org/doc/41991},

volume = {22},

year = {2006},

}

TY - JOUR

AU - Saloff-Coste, Laurent

AU - Woess, Wolfgang

TI - Transition operators on co-compact G-spaces.

JO - Revista Matemática Iberoamericana

PY - 2006

VL - 22

IS - 3

SP - 747

EP - 799

AB - 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 Lp-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.

LA - eng

KW - Cadenas de Markov; Probabilidad de transición; Operador laplaciano; G-espacios; Invariancia; -space; proper action; isometries; invariant transition operator; amenability

UR - http://eudml.org/doc/41991

ER -

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