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Groups and Surfaces

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If you have a question about this talk, please contact Simon Goodwin.

A map on a compact surface can be described as a transitive finite permutation representation of a triangle group. There is a natural complex structure on the underlying surface of the map, making it a Riemann surface, or equivalently a complex algebraic curve. Belyi’s Theorem states that the algebraic curves arising from maps are those defined over algebraic number fields, giving a faithful action of the absolute Galois group (the Galois group of the field of algebraic numbers) on maps. This motivates efforts to classify maps, especially in the regular (most symmetric) case, and to understand the action of the absolute Galois group on maps. I shall illustrate this in the case of the Fermat curves and their generalisations, where one can apply work of Huppert, Ito and Wielandt on groups factorising as products of cyclic groups, and of Hall on solvable groups. If there is time I will mention recent work on Beauville surfaces: these are complex algebraic surfaces with certain rigidity properties, obtained from finite groups acting on pairs of regular maps; it is conjectured that every non-abelian finite simple group except A5 can be used here.

This talk is part of the Algebra seminar series.

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