On procongruence curve complexes and their automorphisms

This paper launches the exploration of the procongruence completions for 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 the Grothendieck–Teichmüller theory. Here several results of foundational nature are stated and proved, among which reconstruction theorems in the discrete and complete settings, which give a graph theoretic characterization 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–Teichmüller group, to be studied elsewhere. 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.