Yet another p-adic hyperbolic disc

TL;DR

This paper introduces new p-adic hyperbolic discs in the projective p-adic plane, defining a p-adic convexity and Hilbert distance, and explores their geometric properties and relation to p-adic trees.

Contribution

It constructs novel p-adic hyperbolic discs with a geometric approach, defines p-adic convexity and Hilbert distance, and analyzes their symmetry groups and relation to p-adic trees.

Findings

Existence of three p-adic hyperbolic discs for odd p and seven for p=2.

Only PGL(2,Q_p) acts as isometries preserving the convex structure.

Constructed a quasi-isometric projection from the discs to the p-adic tree.

Abstract

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). Let us stress out that neither the usual notion of convexity nor that of connectedness as known for the real case are meaningful in the p-adic case. us, there will be a rephrasing game for the denitions of real convexity until we reach a formulation suitable for other local elds. It will lead us to a denition of p-adic convexity by duality. Although we will not recover the beautiful behaviour of real convexity, we will still be able to dene the most important tool for our goals, namely the Hilbert distance. We construct our analogues of the hyperbolic disc (once again, via the projective model of the hyperbolic plane) in a quite geometric, even naive, way. Our construction gives 2-dimensional objects over Q p. It is very dierent, in spirit and in facts, of Drinfeld p-adic hyperbolic plane []. e possible relations between the two objects remain still unexplored. Another object often viewed as an analogue of the hyperbolic disc is the tree of PGL.2; Q p / []. We explore the relations between our discs and this tree, constructing a natural quasi-isometric projection from the discs to the tree. Eventually we explore the transformation groups of our discs. And, whereas the transformation group of the tree is huge, we prove that only PGL.2; Q p / acts on the discs preserving the convex structure.