Unlikely intersections problem for automorphisms of Markov surfaces
Vortrag von Dr. Marc Abboud
Datum: 05.03.25 Zeit: 13.30 - 14.30 Raum: ETH HG G 19.1
The family of Markov surfaces M_D is a family of cubic affine surfaces parametrised by the complex line (here D is the parameter) that is related to the character variety of the punctured torus. It has sparked a lot of work in algebraic geometry, algebraic differential equations and low-dimensional topology. The mapping class group of the punctured torus acts on the character variety and preserves this family. Its image is a finite index subgroup of the group of automorphisms of M_D for every parameter D. We show the following result: for certain algebraic parameters D if two automorphisms of M_D with positive entropy share a Zariski dense set of periodic points then they share a common iterate. The proof uses tools from arithmetic dynamics such as the arithmetic equidistribution theorem of Yuan and Zhang and tools from quasi-Fuchsian representation theory.