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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.
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