ON THE PROCONGRUENCE COMPLETION OF THE TEICHMÜLLER MODULAR GROUP.

For 2g - 2 + n > 0, the Teichmüller modular group Γg,n of a compact Riemann surface of genus g with n points removed Sg,n is the group of homotopy classes of diffeomorphisms of Sg,n which preserve the orientation of Sg,n and a given order of its punctures. Let Γg,n be the fundamental group of Sg,n, with a given base point, and ... its profinite completion. There is then a natural faithful representation Γg,n ↪ Out(...). The procongruence completion ... of the Teichmüller group is defined to be the closure of the Teichmüller group Γg,n inside the profinite group Out(...). In this paper, we begin a systematic study of the procongruence completion .... The set of profinite Dehn twists of ... is the closure, inside this group, of the set of Dehn twists of Γg,n. The main technical result of the paper is a parametrization of the set of profinite Dehn twists of ... and the subsequent description of their centralizers. This is the basis for the Grothendieck-Teichmüller Lego with procongruence Teichmüller groups as building blocks. As an application, in Section 7, we prove that some Galois representations associated to hyperbolic curves over number fields and their moduli spaces are faithful. [ABSTRACT FROM AUTHOR]