Faithful actions of the absolute Galois group on connected components of moduli spaces
Abstract
We give a canonical procedure associating to an algebraic number a first a hyperelliptic curve Ca, and then a triangle curve (Da, Ga) obtained through the normal closure of an associated Belyi function. In this way we show that the absolute Galois group Gal( /) acts faithfully on the set of isomorphism classes of marked triangle curves, and on the set of connected components of marked moduli spaces of surfaces isogenous to a higher product (these are the free quotients of a product C1 x C2 of curves of respective genera g1, g2 >= 2 by the action of a finite group G). We show then, using again the surfaces isogenous to a product, first that it acts faithfully on the set of connected components of moduli spaces of surfaces of general type (amending an incorrect proof in a previous ArXiv version of the paper); and then, as a consequence, we obtain that for every element σ ∈ ( /), not in the conjugacy class of complex conjugation, there exists a surface of general type X such that X and the Galois conjugate surface Xσ have nonisomorphic fundamental groups. Using polynomials with only two critical values, we can moreover exhibit infinitely many explicit examples of such a situation.