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Analogies between real hyperbolic spaces and Bruhat- Tits trees over \(\mathbb{Q}_ p\). (Analogies entre espaces hyperboliques réels et l’arbre de Bruhat-Tits sur \(\mathbb{Q}{}_ p\).) (French) Zbl 0747.51010

Sémin. Théor. Spectrale Géom., Chambéry-Grenoble 9, Année 1990-1991, 95-101 (1991).
The author gives an elementary account of the well-known analogy between real hyperbolic 2- and 3- space, viewed as symmetric space for the group \(SL(2,\mathbb{R})\) resp. \(SL(S,\mathbb{C})\), and the Bruhat-Tits tree associated with the group \(SL(2,\mathbb{Q}_ P)\). Using this analogy, he solves the Dirichlet problem and derives an explicit Poisson integral formula for harmonic functions defined on the aforementioned tree.

MSC:

51M10 Hyperbolic and elliptic geometries (general) and generalizations
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31B25 Boundary behavior of harmonic functions in higher dimensions
05C05 Trees
Full Text: EuDML