|Seung uk Jang
|University of Chicago
Non-archimedean fields and varieties over them admit the operation of tropicalizations, which provides a piecewise-linear approximate sketch of varieties that encapsulates many key aspects. For Markov surfaces $x^2+y^2+z^2+xyz=D$, this viewpoint was initiated by works of Spalding and Veselov, who focused on its tropical and dynamical aspects.
In this talk, we will be working on a more general family of Markov surfaces and discover that, for any parameters, we have a copy or a shadow of the hyperbolic plane with the $(infty,infty,infty)$-triangle reflection group action. Such a viewpoint easily yields corollaries in Fatou domains (dynamical side) or the finiteness of orbits of rational points with prime power denominators (number theory side). Some interesting number-theoretic aspects of this system may be introduced, such as Farey triples.