Geodesic
Yet another $p$-adic hyperbolic disc: Hilbert distance for $p$-adic fields
Antonin Guilloux1 orcid
1Université Pierre et Marie Curie, Paris, France
Groups, geometry, and dynamics, Tome 10 (2016) no. 1, pp. 9-43

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DOI

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 P2​(Qp​) naturally endowed with a "Hilbert" distance and a transitive action of PGL(2, Qp​) by isometries. These 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 find three of them if p is odd (and seven if p=2).
DOI : 10.4171/ggd/341
Classification : 52-XX
Mots-clés : Hilbert distance, hyperbolic disc, local fields

Antonin Guilloux  1

1 Université Pierre et Marie Curie, Paris, France 
Antonin Guilloux. Yet another $p$-adic hyperbolic disc: Hilbert distance for $p$-adic fields. Groups, geometry, and dynamics, Tome 10 (2016) no. 1, pp. 9-43. doi: 10.4171/ggd/341
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     journal = {Groups, geometry, and dynamics},
     pages = {9--43},
     year = {2016},
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     number = {1},
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     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/341/}
}
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