[SNU Number Theory Seminar 30 June] Do Tropical Markov Cubics dream of Hyperbolic Origami?

Place:  27-325 (SNU)

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.