Markoff surfaces and strong approximation 
Abstract: We discuss the transitivity properties of the group of morphisms generated by Vieta involutions on the solutions in congruences to the Markoff equation as well as to other Markoff type affine cubic surfaces. These are dictated by the finite orbits of these actions on the algebraic points. The latter can be determined effectively and in special cases is connected to the problem of determining all algebraic Painleve VI's. Applications to forms of strong approximation for integer points and to sieving on such affine surfaces, as well as to Markoff numbers will be given. Joint work with J.Bourgain and A.Gamburd. 

Web page: Alexandru I. Suciu  Comments to: i.loseu@neu.edu  
Posted: November 5, 2015  URL: http://www.northeastern.edu/iloseu/bhmn/sarnak15.html 