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We describe in this paper a geometric construction in the projective p-adic plane that gives, together with a suitable notion of p-adic convexity, some open subsets of P 2 .Q p / naturally endowed with a "Hilbert" distance and a transitive action of PGL.2; Q p / by isometries.
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Jean-Pierre Serre, Trees , Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003, Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation
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Constantin Vernicos, Introduction aux géométries de Hilbert , Actes de Séminaire de Théorie Spectrale et Géométrie. Vol. 23. Année 2004–2005, Sémin. Théor. Spectr. Géom., vol. 23, Univ. Grenoble I, Saint, 2005, pp. 145–168
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Yves Benoist, A survey on divisible convex sets , Geometry, analysis and topology of discrete groups, Adv. Lect. Math. (ALM), vol. 6, Int. Press, Somerville, MA, 2008, pp. 1–18
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