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CATEGORIES:Algebra Seminar
SUMMARY:Groups and Surfaces - Gareth Jones\, University of
Southampton
DTSTART:20091022T150000Z
DTEND:20091022T160000Z
UID:TALK107AT
URL:/talk/index/107
DESCRIPTION:A map on a compact surface can be described as a t
ransitive finite permutation representation of a t
riangle group. There is a natural complex structur
e on the underlying surface of the map\, making it
a Riemann surface\, or equivalently a complex alg
ebraic curve. Belyi's Theorem states that the alge
braic curves arising from maps are those defined o
ver algebraic number fields\, giving a faithful ac
tion of the absolute Galois group (the Galois grou
p of the field of algebraic numbers) on maps. This
motivates efforts to classify maps\, especially i
n the regular (most symmetric) case\, and to under
stand the action of the absolute Galois group on m
aps. I shall illustrate this in the case of the Fe
rmat curves and their generalisations\, where one
can apply work of Huppert\, Ito and Wielandt on gr
oups factorising as products of cyclic groups\, an
d of Hall on solvable groups. If there is time I w
ill mention recent work on Beauville surfaces: the
se are complex algebraic surfaces with certain rig
idity properties\, obtained from finite groups act
ing on pairs of regular maps\; it is conjectured t
hat every non-abelian finite simple group except _
A__{5} can be used here.
LOCATION:Watson Building\, Lecture Room A
CONTACT:Simon Goodwin
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