Some spectral and geometric aspects of countable groups

Alexander Bendikov (1) , Barbara Bobikau (1) , Christophe Pittet (2)

1 Mathematical Institute, University of Wroclaw
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités

We discuss the relationship between the isospectral profile and the spectral distribution of a Laplace operator on a countable group. In the case of locally finite countable groups, we emphasize the relevance of the metric associated to a natural Markov operator: it is an ultra-metric whose balls are optimal sets for the isospectral profile.

Mots clés: probability measures on graphs and groups; geometric group theory; spectral theory and eigenvalue problems for Laplace operators

Domaines: Probabilités [math.PR]; Théorie des groupes [math.GR]

Alexander Bendikov, Barbara Bobikau, Christophe Pittet. Some spectral and geometric aspects of countable groups. Random Walks, Boundaries and Spectra, 64, Springer Basel AG, pp.227-234, 2011, Progress in Probability, 978-3-0346-0243-3. ⟨10.1007/978-3-0346-0244-0_12⟩. ⟨hal-01305041⟩