On the geometric outer monodromy representation and the Grothendieck conjecture of hyperbolic curves

Let \(l\) be a prime number. The étale fundamental group of the moduli stack of hyperbolic curves over the field of complex numbers has a natural outer action on the étale fundamental group of a hyperbolic curve over the field of complex numbers. This outer action is called the geometric outer monodromy representation. Since the geometric outer monodromy representation is induced by the profinite completion of the mapping class group, the geometric outer monodromy representation may be regarded as a topological object. On the other hand, there exist hyperbolic curves whose images of pro-\(l\) outer Galois representations contain the image of the geometric pro-\(l\) outer monodromy representation. Such a hyperbolic curve is called \(l\)-monodromically full. By the definition, it may be expected that an \(l\)-monodromically full hyperbolic curve is studied by way of the geometric pro-\(l\) outer monodromy representation. In this talk, I recall well-known results concerning the geometric outer monodromy representation and \(l\)-monodromically full hyperbolic curves, and explain that the pro-\(l\) Grothendieck conjecture of certain hyperbolic curves over some infinite number fields follows from these results.