May 27 – 31, 2024
Institut Henri Poincaré
Europe/Paris timezone
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Markoff Surfaces in the p-adic World

May 30, 2024, 11:00 AM
Amphithéâtre Hermite / Darboux (Institut Henri Poincaré)

Amphithéâtre Hermite / Darboux

Institut Henri Poincaré

11 rue Pierre et Marie Curie 75005 Paris


Seung uk Jang (Université de Rennes)


Markoff surfaces appear in studies of the character variety of the
$1$-punctured torus or the $4$-punctured sphere, which have many algebraic
automorphisms. When we sketch their real points, say, we often observe
'hyperbolic' and "spherical'' parts. The dynamical nature of the
algebraic automorphisms on these respective parts is well-known for real
(or complex) points.

In the talk, we will discuss what happens when we ask an analogous
question for $p$-adic numbers. It turns out that (a) the tropicalization
of the variety gives rise to a copy of the hyperbolic plane, and (b)
there is a finite list of bounded, automorphism-invariant closed subsets
over $p$-adic points. These correspond to the behaviors of "hyperbolic"
and "spherical'' parts in the $p$-adic case.

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