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Yet another p-adic hyperbolic disc · Around

2016

Yet another p-adic hyperbolic disc

Guilloux, Antonin

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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.

ese open sets are natural analogues of the hyperbolic disc, more precisely of Klein's projective model.
But, unlike the real case, there is not only one such hyperbolic disc.
Indeed, we nd three of them if p is odd (and seven if p D 2).

Built on

O. Goldman and N. Iwahori, The space of 𝔭 {\mathfrak{p}} -adic norms , Acta Math. 109
1963
Earlier work this paper cites.
O. Timothy O’Meara, Introduction to quadratic forms , Classics in Mathematics, Springer-Verlag, Berlin, 2000, Reprint of the 1973 edition
1973
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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
1980
Earlier work this paper cites.

Similar

J.-F. Boutot and H. Carayol, Uniformisation p p -adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfel ′
1988
Cited alongside, same era.
Pierre de la Harpe, On Hilbert’s metric for simplices , Geometric group theory, Vol. 1 (Sussex, 1991), London Math. Soc. Lecture Note Ser., vol. 181, Cambridge Univ. Press, Cambridge, 1993, pp. 97–119
1993
Cited alongside, same era.
A. F. Beardon, The Klein, Hilbert and Poincaré metrics of a domain , J. Comput. Appl. Math. 105
1997
Cited alongside, same era.
D. Hilbert, Ueber die gerade Linie als kürzeste Verbindung zweier Punkte , Math. Ann. 46
Cited in the paper.

Then

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
2005
Later among the works it cites.
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
2008
Later among the works it cites.

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