"Moscow--Petersburg"
seminar on low-dimensional mathematics
SPb, PDMI, 28.09.2001
S.K.Lando (IUM, Moscow) "Galois group acting on graphs"
The action of the Galois group Gal(\bar{Q}/Q) on graphs embedded
in two-dimensional surfaces, which was discovered by A.Grothendieck,
relies on two facts:
1. (topological, known for about 100 years) -- a ramified covering
of the sphere corresponds to a graph embedded in the covering
surface, and this correspondence is in a sense one-to-one;
2. (algebro-geometric, a recent theorem of G.Belyi) -- a complex
curve is defined over the field of algebraic numbers if and only if
it carries a meromorphic function with no more than 3 critical
values. The natural action of the Galois group on the set of
meromorphic functions defined over the field of algebraic numbers,
defines its action on embedded graphs.
In this talk, we will introduce the basic notions and discuss
certain invariants of this action. Numerous examples will be given.
Home page of the seminar: http://www.pdmi.ras.ru/~duzhin/lowdimma.html