ON PROCONGRUENCE CURVE COMPLEXES AND THEIR AUTOMORPHISMS

In this paper we start exploring the procongruence completions of three varieties of curve complexes attached to hyperbolic surfaces, as well as their automorphisms groups. The discrete counterparts of these objects, especially the curve complex and the so-called pants complex were defined long ago and have been the subject of numerous studies. Introducing some form of completions is natural and indeed necessary to lay the ground for a topological version of Grothendieck-Teichmuller theory. Based on previous work by the first author, we state and prove several basic results, among which reconstruction theorems in the discrete and complete settings, which give a graph theoretic characterizations of versions of the curve complex as well as a rigidity theorem for the complete pants complex, in sharp contrast with the case of the (complete) curve complex, whose automorphisms actually define a version of the Grothendieck-Teichmuller group, to be studied elsewhere (see [22]). We also prove an anabelian theorem pertaining to the moduli stacks of curves, one of the very few such results available in higher dimensions. We work all along with the procongruence completions-and for good reasons-recalling however that the so-called congruence conjecture predicts that this completion should coincide with the full profinite completion.