2016

Yet another p-adic hyperbolic disc

Guilloux, Antonin

Understand

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

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Similar

  • J.-F. Boutot and H. Carayol, Uniformisation p p -adique des courbes de Shimura: les thĂ©orèmes de ÄŚerednik et de Drinfel ′

    1988

    Cited alongside, same era.

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    Cited alongside, same era.

  • A. F. Beardon, The Klein, Hilbert and PoincarĂ© metrics of a domain , J. Comput. Appl. Math. 105

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    Cited alongside, same era.

  • D. Hilbert, Ueber die gerade Linie als kĂĽrzeste Verbindung zweier Punkte , Math. Ann. 46

    Cited in the paper.

Then

  • Constantin Vernicos, Introduction aux gĂ©omĂ©tries de Hilbert , Actes de SĂ©minaire de ThĂ©orie Spectrale et GĂ©omĂ©trie. Vol. 23. AnnĂ©e 2004–2005, SĂ©min. ThĂ©or. Spectr. GĂ©om., 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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