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Non-planarity of Markoff graphs mod p

Algebra & Discrete Mathematics

Speaker: Matthew de Courcy-Ireland, Stockholm University
Location: 2112 MSB
Start time: Fri, Oct 14 2022, 3:10PM

We discuss a family of cubic surfaces defined by x^2+y^2+z^2=xyz+k modulo prime numbers. The solutions form a graph, where each vertex (x,y,z) is joined to the other solution of the same quadratic in any of the three variables. These moves are related to a nonlinear action of the modular group PGL(2,Z) on the surface. We outline some ways these equations arise in geometry, and how we became interested in showing that the associated graphs cannot be embedded in the plane. We sketch a proof that 2, 3, and 7 are the only primes for which the graph is planar. For larger primes, we give an estimate of the Euler characteristic of a surface in which the graph can be embedded.

Will also be streamed on Zoom (