Pure Colloquium Seminar - The Farey Maps Modulo n, David Singerman, Southampton Seminar

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The Farey graph is a graph whose vertices are the rational numbers together with 1/0=infinity, and where a/c is joined by an edge to b/d if ad-bc=1 or -1. This gives us a triangular tessellation (or map) on the closed upper-half plane. If we regard all our integers modulo n, for some positive integer n we get finite maps which are often very interesting. For example if n=5 we get the icosahedron, and for n=7 we get Klein’s map on his famous Riemann surface of genus 3. Thus we are using elementary arithmetic to construct interesting regular maps and Riemann surfaces. This is joint work with James Strudwick.